continuous cohomology and homology of profinite groups · keywords and phrases: continuous...
Embed Size (px)
TRANSCRIPT
Documenta Math 1269
Continuous Cohomology and Homology
of Profinite Groups
Marco Boggi and Ged Corob Cook
Received April 29 2015
Revised July 9 2016
Communicated by Otmar Venjakob
Abstract We develop cohomological and homological theories for aprofinite groupG with coefficients in the Pontryagin dual categories ofprodiscrete and indprofinite Gmodules respectively The standardresults of group (co)homology hold for this theory we prove versionsof the Universal Coefficient Theorem the LyndonHochschildSerrespectral sequence and Shapirorsquos Lemma
2010 Mathematics Subject Classification Primary 20J06 Secondary20E18 20J05 13J10Keywords and Phrases Continuous cohomology profinite groupsquasiabelian categories
Introduction
Cohomology groups Hn(GM) can be studied for profinite groups G in muchthe same way as abstract groups The coefficients M will lie in some categoryof topological modules but it is not clear what the right category is Theclassical solution is to allow only discrete modules in which case Hn(GM)is discrete see [9] for this approach For many applications it is useful totake M to be a profinite Gmodule A cohomology theory allowing discreteand profinite coefficients is developed in [12] when G is of type FPinfin but forarbitrary profinite groups there has not previously been a satisfactory definitionof cohomology with profinite coefficients A difficulty is that the category ofprofinite Gmodules does not have enough injectivesWe define the cohomology of a profinite group with coefficients in the categoryof prodiscrete ZJGKmodules PD(ZJGK) This category contains the discrete
Documenta Mathematica 21 (2016) 1269ndash1312
1270 Marco Boggi and Ged Corob Cook
ZJGKmodules and the secondcountable profinite ZJGKmodules when G itselfis secondcountable this is sufficient for many applicationsPD(ZJGK) is not an abelian category instead it is quasiabelian ndash homologicalalgebra over this generalisation is treated in detail in [8] and [10] and we givean overview of the results we will need in Section 5 Working over the derivedcategory this allows us to define derived functors and study their propertiesthese functors exist because PD(ZJGK) has enough injectives The resultingcohomology theory does not take values in a module category but rather inthe heart of a canonical tstructure on the derived category RH(PD(Z)) in
which PD(Z) is a coreflective subcategoryThe most important result of this theory is that it allows us to prove a LyndonHochschildSerre spectral sequence for profinite groups with profinite coefficients This has not been possible in previous formulations of profinite cohomology and should allow the application of a wide range of techniques fromabstract group cohomology to the study of profinite groups A good example ofthis is the use of the spectral sequence to give a partial answer to a conjectureof Krophollerrsquos [9 Open Question 6121] in a paper by the second author [3]We also define a homology theory for profinite groups which extends the category of coefficient modules to the indprofinite Gmodules As in previousexpositions this is entirely dual to the cohomology theoryFinally in Section 8 we compare this theory to previous cohomology theoriesfor profinite groups It is naturally isomorphic to the classical cohomology ofprofinite groups with discrete coefficients and to the SymondsWeigel theoryfor profinite modules of type FPinfin with profinite coefficients We also definea continuous cochain cohomology constructed by considering only the continuous Gmaps from the standard bar resolution of a topological group G to atopological Gmodule M with the compactopen topology and taking its cohomology the comparison here is more nuanced but we show that in certaincircumstances these cohomology groups can be recovered from oursTo clarify some terminology it is common to refer to groups modules and soon without a topology as discrete However this creates an ambiguity in thissituation For a profinite ring R there are Rmodules M without a topologysuch that giving M the discrete topology creates a topological group on whichthe Raction R times M rarr M is not continuous Therefore a discrete modulewill mean one for which the Raction is continuous and we will call algebraicobjects without a topology abstract
1 IndProfinite Modules
We say a topological space X is indprofinite if there is an injective sequenceof subspaces Xi i isin N whose union is X such that each Xi is profinite andX has the colimit topology with respect to the inclusions Xi rarr X That isX = lim
minusrarrIPSpaceXi We write IPSpace for the category of indprofinite spaces
and continuous maps
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1271
Proposition 11 Given an indprofinite space X defined as the colimit of aninjective sequence Xi of profinite spaces any compact subspace K of X iscontained in some Xi
Proof [5 Proposition 11] proves this under the additional assumption thatthe Xi are profinite groups but the proof does not use this
This shows that compact subspaces of X are exactly the profinite subspacesand that if an indprofinite space X is defined as the colimit of a sequenceXi then the Xi are cofinal in the poset of compact subspaces of X Wecall such a sequence a cofinal sequence for X any cofinal sequence of profinitesubspaces defines X up to homeomorphismA topological space X is called compactly generated if it satisfies the followingcondition a subspace U of X is closed if and only if U cap K is closed in Kfor every compact subspace K of X See [11] for background on such spacesBy the definition of the colimit topology indprofinite spaces are compactlygenerated Indeed a subspace U of an indprofinite space X is closed if andonly if U capXi is closed in Xi for all i if and only if U capK is closed in K forevery compact subspace K of X by Proposition 11
Lemma 12 IPSpace has finite products and coproducts
Proof Given XY isin IPSpace with cofinal sequences Xi Yi we can construct X ⊔ Y using the cofinal sequence Xi ⊔ Yi However it is not clearwhether X times Y with the product topology is indprofinite Instead thanks toProposition 11 the indprofinite space lim
minusrarrXi times Yi is the product of X and Y
it is easy to check that it satisfies the relevant universal property
Moreover by the proposition Xi times Yi is cofinal in the poset of compactsubspaces of X times Y (with the product topology) and hence lim
minusrarrXi times Yi is the
kification of X times Y or in other words it is the product of X and Y in thecategory of compactly generated spaces ndash see [11] for details So we will writeX timesk Y for the product in IPSpaceWe say an abelian group M equipped with an indprofinite topology is an indprofinite abelian group if it satisfies the following condition there is an injectivesequence of profinite subgroups Mi i isin N which is a cofinal sequence for theunderlying space of M It is easy to see that profinite groups and countablediscrete torsion groups are indprofinite Moreover Qp is indprofinite via thecofinal sequence
Zpmiddotpminusrarr Zp
middotpminusrarr middot middot middot (lowast)
Remark 13 It is not obvious that indprofinite abelian groups are topologicalgroups In fact we see below that they are But it is much easier to seethat they are kgroups in the sense of [7] the multiplication map M timesk M =limminusrarrIPSpace
Mi times Mi rarr M is continuous by the definition of colimits The
kgroup intuition will often be more useful
Documenta Mathematica 21 (2016) 1269ndash1312
1272 Marco Boggi and Ged Corob Cook
In the terminology of [5] the indprofinite abelian groups are just the abelianweakly profinite groups We recall some of the basic results of [5]
Proposition 14 Suppose M is an indprofinite abelian group with cofinalsequence Mi
(i) Any compact subspace of M is contained in some Mi
(ii) Closed subgroups N of M are indprofinite with cofinal sequence N capMi
(iii) Quotients of M by closed subgroups N are indprofinite with cofinal sequence Mi(N capMi)
(iv) Indprofinite abelian groups are topological groups
Proof [5 Proposition 11 Proposition 12 Proposition 15]
As before we call a sequence Mi of profinite subgroups making M into anindprofinite group a cofinal sequence for M Suppose from now on that R is a commutative profinite ring and Λ is a profiniteRalgebra
Remark 15 We could define indprofinite rings as colimits of injective sequences (indexed by N) of profinite rings and much of what follows does holdin some sense for such rings but not much is lost by the restriction In particular it would be nice to use the machinery of indprofinite rings to study Qpbut the sequence (lowast) making Qp into an indprofinite abelian group does notmake it into an indprofinite ring because the maps are not maps of rings
We say that M is a left Λkmodule if M is a kgroup equipped with a continuous map ΛtimeskM rarrM A Λkmodule homomorphism M rarr N is a continuousmap which is a homomorphism of the underlying abstract Λmodules BecauseΛ is profinite Λtimesk M = ΛtimesM so ΛtimesM rarrM is continuous Hence if M isa topological group (that is if multiplication M timesM rarrM is continuous) thenit is a topological ΛmoduleWe say that a left Λkmodule M equipped with an indprofinite topology isa left indprofinite Λmodule if there is an injective sequence of profinite submodules Mi i isin N which is a cofinal sequence for the underlying space of M So countable discrete Λmodules are indprofinite because finitely generateddiscrete Λmodules are finite and so are profinite Λmodules In particular Λwith leftmultiplication is an indprofinite Λmodule Note that since profiniteZmodules are the same as profinite abelian groups indprofinite Zmodules arethe same as indprofinite abelian groupsThen we immediately get the following
Corollary 16 Suppose M is an indprofinite Λmodule with cofinal sequenceMi
(i) Any compact subspace of M is contained in some Mi
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1273
(ii) Closed submodules N of M are indprofinite with cofinal sequence NcapMi
(iii) Quotients of M by closed submodules N are indprofinite with cofinalsequence Mi(N capMi)
(iv) Indprofinite Λmodules are topological Λmodules
As before we call a sequence Mi of profinite submodules making M into anindprofinite Λmodule a cofinal sequence for M
Lemma 17 Indprofinite Λmodules have a fundamental system of neighbourhoods of 0 consisting of open submodules Hence such modules are Hausdorffand totally disconnected
Proof Suppose M has cofinal sequence Mi and suppose U subeM is open with0 isin U by definition U capMi is open in Mi for all i Profinite modules havea fundamental system of neighbourhoods of 0 consisting of open submodulesby [9 Lemma 511] so we can pick an open submodule N0 of M0 such thatN0 sube U capM0 Now we proceed inductively given an open submodule Ni of Mi
such that Ni sube U capMi let f be the quotient map M rarrMNi Then f(U) isopen in MNi by [5 Proposition 13] so f(U)capMi+1Ni is open in Mi+1NiPick an open submodule of Mi+1Ni which is contained in f(U) capMi+1Ni
and write Ni+1 for its preimage in Mi+1 Finally let N be the submodule ofM with cofinal sequence Ni N is open and N sube U as required
Write IP (Λ) for the category whose objects are left indprofinite Λmodulesand whose morphisms M rarr N are Λkmodule homomorphisms We willidentify the category of right indprofinite Λmodules with IP (Λop) in the usualway Given M isin IP (Λ) and a submodule M prime write M prime for the closure of M prime
in M Given MN isin IP (Λ) write HomIPΛ (MN) for the abstract Rmodule
of morphisms M rarr N this makes HomIPΛ (minusminus) into a functor IP (Λ)op times
IP (Λ)rarrMod(R) in the usual way where Mod(R) is the category of abstractRmodules and Rmodule homomorphisms
Proposition 18 IP (Λ) is an additive category with kernels and cokernels
Proof The category is clearly preadditive the biproduct MoplusN is the biproduct of the underlying abstract modules with the topology of M timesk N Theexistence of kernels and cokernels follows from Corollary 16 the cokernel off M rarr N is Nf(M)
Remark 19 The category IP (Λ) is not abelian in general Consider the countable direct sum oplusalefsym0
Z2Z with the discrete topology and the countable direct product
prodalefsym0
Z2Z with the profinite topology Both are indprofinite
Zmodules There is a canonical injective map i oplusZ2Z rarrprod
Z2Z buti(oplusZ2Z) is not closed in
prodZ2Z Moreover oplusZ2Z is not homeomorphic to
i(oplusZ2Z) with the subspace topology because i(oplusZ2Z) is not discrete bythe construction of the product topology
Documenta Mathematica 21 (2016) 1269ndash1312
1274 Marco Boggi and Ged Corob Cook
Given a morphism f M rarr N in a category with kernels and cokernelswe write coim(f) for coker(ker(f)) and im(f) for ker(coker(f)) That iscoim(f) = f(M) with the quotient topology coming from M and im(f) =f(M) with the subspace topology coming from N In an abelian categorycoim(f) = im(f) but the preceding remark shows that this fails in IP (Λ)We say a morphism f M rarr N in IP (Λ) is strict if coim(f) = im(f) Inparticular strict epimorphisms are surjections Note that if M is profinite allmorphisms f M rarr N must be strict because compact subspaces of Hausdorffspaces are closed so that coim(f)rarr im(f) is a continuous bijection of compactHausdorff spaces and hence a topological isomorphism
Proposition 110 Morphisms f M rarr N in IP (Λ) such that f(M) is aclosed subset of N have continuous sections So f is strict in this case and inparticular continuous bijections are isomorphisms
Proof [5 Proposition 16]
Corollary 111 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if f is strict
Proof The decomposition is the usual one M rarr coim(f)gminusrarr im(f) rarr N
for categories with kernels and cokernels Clearly coim(f) = f(M) rarr N isinjective so g is too and hence g is monic Also the settheoretic image ofM rarr im(f) is dense so the settheoretic image of g is too and hence g is epicThen everything follows from Proposition 110
Because IP (Λ) is not abelian it is not obvious what the right notion of exactness is We will say that a chain complex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M Despite the failure of our category to be abelian we can prove the followingSnake Lemma which will be useful later
Lemma 112 Suppose we have a commutative diagram in IP (Λ) of the form
L
f
Mp
g
N
h
0
0 Lprime i M prime N prime
such that the rows are strict exact at MNLprimeM prime and f g h are strict Thenwe have a strict exact sequence
ker(f)rarr ker(g)rarr ker(h)partminusrarr coker(f)rarr coker(g)rarr coker(h)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1275
Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forgetting the topology and working with abstract Λmodules we get the sequencedescribed above from the standard Snake Lemma for abstract modules whichis exact as a sequence of abstract modules This implies that if all the mapsin the sequence are continuous then they have closed settheoretic image andhence the sequence is strict by Proposition 110 To see that part is continuouswe construct it as a composite of continuous maps Since coim(p) = N byProposition 110 again p has a continuous section s1 N rarr M and similarlyi has a continuous section s2 im(i) rarr Lprime Then as usual part = s2gs1 Thecontinuity of the other maps is clear
Proposition 113 The category IP (Λ) has countable colimits
Proof We show first that IP (Λ) has countable direct sums Given a countablecollection Mn n isin N of indprofinite Λmodules write Mni i isin N foreach n for a cofinal sequence for Mn Now consider the injective sequenceNn given by Nn =
prodni=1 Min+1minusi each Nn is a profinite Λmodule so
the sequence defines an indprofinite Λmodule N It is easy to check that theunderlying abstract module of N is
oplusn Mn that each canonical map Mn rarr N
is continuous and that any collection of continuous homomorphisms Mn rarr Pin IP (Λ) induces a continuous N rarr P
Now suppose we have a countable diagram Mn in IP (Λ) Write S for theclosed submodule of
oplusMn generated (topologically) by the elements with jth
component minusx kth component f(x) and all other components 0 for all mapsf Mj rarr Mk in the diagram and all x isin Mj By standard arguments(oplus
Mn)S with the quotient topology is the colimit of the diagram
Remark 114 We get from this construction that given a countable collectionof short strict exact sequences
0rarr Ln rarrMn rarr Nn rarr 0
in IP (Λ) their direct sum
0rarroplus
Ln rarroplus
Mn rarroplus
Nn rarr 0
is strict exact by Proposition 110 because the sequence of underlying modulesis exact So direct sums preserve kernels and cokernels and in particular directsums preserve strict maps because given a countable collection of strict mapsfn in IP (Λ)
coim(oplus
fn) = coker(ker(oplus
fn)) =oplus
coker(ker(fn))
=oplus
ker(coker(fn)) = ker(coker(oplus
fn)) = im(oplus
fn)
Documenta Mathematica 21 (2016) 1269ndash1312
1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

1270 Marco Boggi and Ged Corob Cook
ZJGKmodules and the secondcountable profinite ZJGKmodules when G itselfis secondcountable this is sufficient for many applicationsPD(ZJGK) is not an abelian category instead it is quasiabelian ndash homologicalalgebra over this generalisation is treated in detail in [8] and [10] and we givean overview of the results we will need in Section 5 Working over the derivedcategory this allows us to define derived functors and study their propertiesthese functors exist because PD(ZJGK) has enough injectives The resultingcohomology theory does not take values in a module category but rather inthe heart of a canonical tstructure on the derived category RH(PD(Z)) in
which PD(Z) is a coreflective subcategoryThe most important result of this theory is that it allows us to prove a LyndonHochschildSerre spectral sequence for profinite groups with profinite coefficients This has not been possible in previous formulations of profinite cohomology and should allow the application of a wide range of techniques fromabstract group cohomology to the study of profinite groups A good example ofthis is the use of the spectral sequence to give a partial answer to a conjectureof Krophollerrsquos [9 Open Question 6121] in a paper by the second author [3]We also define a homology theory for profinite groups which extends the category of coefficient modules to the indprofinite Gmodules As in previousexpositions this is entirely dual to the cohomology theoryFinally in Section 8 we compare this theory to previous cohomology theoriesfor profinite groups It is naturally isomorphic to the classical cohomology ofprofinite groups with discrete coefficients and to the SymondsWeigel theoryfor profinite modules of type FPinfin with profinite coefficients We also definea continuous cochain cohomology constructed by considering only the continuous Gmaps from the standard bar resolution of a topological group G to atopological Gmodule M with the compactopen topology and taking its cohomology the comparison here is more nuanced but we show that in certaincircumstances these cohomology groups can be recovered from oursTo clarify some terminology it is common to refer to groups modules and soon without a topology as discrete However this creates an ambiguity in thissituation For a profinite ring R there are Rmodules M without a topologysuch that giving M the discrete topology creates a topological group on whichthe Raction R times M rarr M is not continuous Therefore a discrete modulewill mean one for which the Raction is continuous and we will call algebraicobjects without a topology abstract
1 IndProfinite Modules
We say a topological space X is indprofinite if there is an injective sequenceof subspaces Xi i isin N whose union is X such that each Xi is profinite andX has the colimit topology with respect to the inclusions Xi rarr X That isX = lim
minusrarrIPSpaceXi We write IPSpace for the category of indprofinite spaces
and continuous maps
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1271
Proposition 11 Given an indprofinite space X defined as the colimit of aninjective sequence Xi of profinite spaces any compact subspace K of X iscontained in some Xi
Proof [5 Proposition 11] proves this under the additional assumption thatthe Xi are profinite groups but the proof does not use this
This shows that compact subspaces of X are exactly the profinite subspacesand that if an indprofinite space X is defined as the colimit of a sequenceXi then the Xi are cofinal in the poset of compact subspaces of X Wecall such a sequence a cofinal sequence for X any cofinal sequence of profinitesubspaces defines X up to homeomorphismA topological space X is called compactly generated if it satisfies the followingcondition a subspace U of X is closed if and only if U cap K is closed in Kfor every compact subspace K of X See [11] for background on such spacesBy the definition of the colimit topology indprofinite spaces are compactlygenerated Indeed a subspace U of an indprofinite space X is closed if andonly if U capXi is closed in Xi for all i if and only if U capK is closed in K forevery compact subspace K of X by Proposition 11
Lemma 12 IPSpace has finite products and coproducts
Proof Given XY isin IPSpace with cofinal sequences Xi Yi we can construct X ⊔ Y using the cofinal sequence Xi ⊔ Yi However it is not clearwhether X times Y with the product topology is indprofinite Instead thanks toProposition 11 the indprofinite space lim
minusrarrXi times Yi is the product of X and Y
it is easy to check that it satisfies the relevant universal property
Moreover by the proposition Xi times Yi is cofinal in the poset of compactsubspaces of X times Y (with the product topology) and hence lim
minusrarrXi times Yi is the
kification of X times Y or in other words it is the product of X and Y in thecategory of compactly generated spaces ndash see [11] for details So we will writeX timesk Y for the product in IPSpaceWe say an abelian group M equipped with an indprofinite topology is an indprofinite abelian group if it satisfies the following condition there is an injectivesequence of profinite subgroups Mi i isin N which is a cofinal sequence for theunderlying space of M It is easy to see that profinite groups and countablediscrete torsion groups are indprofinite Moreover Qp is indprofinite via thecofinal sequence
Zpmiddotpminusrarr Zp
middotpminusrarr middot middot middot (lowast)
Remark 13 It is not obvious that indprofinite abelian groups are topologicalgroups In fact we see below that they are But it is much easier to seethat they are kgroups in the sense of [7] the multiplication map M timesk M =limminusrarrIPSpace
Mi times Mi rarr M is continuous by the definition of colimits The
kgroup intuition will often be more useful
Documenta Mathematica 21 (2016) 1269ndash1312
1272 Marco Boggi and Ged Corob Cook
In the terminology of [5] the indprofinite abelian groups are just the abelianweakly profinite groups We recall some of the basic results of [5]
Proposition 14 Suppose M is an indprofinite abelian group with cofinalsequence Mi
(i) Any compact subspace of M is contained in some Mi
(ii) Closed subgroups N of M are indprofinite with cofinal sequence N capMi
(iii) Quotients of M by closed subgroups N are indprofinite with cofinal sequence Mi(N capMi)
(iv) Indprofinite abelian groups are topological groups
Proof [5 Proposition 11 Proposition 12 Proposition 15]
As before we call a sequence Mi of profinite subgroups making M into anindprofinite group a cofinal sequence for M Suppose from now on that R is a commutative profinite ring and Λ is a profiniteRalgebra
Remark 15 We could define indprofinite rings as colimits of injective sequences (indexed by N) of profinite rings and much of what follows does holdin some sense for such rings but not much is lost by the restriction In particular it would be nice to use the machinery of indprofinite rings to study Qpbut the sequence (lowast) making Qp into an indprofinite abelian group does notmake it into an indprofinite ring because the maps are not maps of rings
We say that M is a left Λkmodule if M is a kgroup equipped with a continuous map ΛtimeskM rarrM A Λkmodule homomorphism M rarr N is a continuousmap which is a homomorphism of the underlying abstract Λmodules BecauseΛ is profinite Λtimesk M = ΛtimesM so ΛtimesM rarrM is continuous Hence if M isa topological group (that is if multiplication M timesM rarrM is continuous) thenit is a topological ΛmoduleWe say that a left Λkmodule M equipped with an indprofinite topology isa left indprofinite Λmodule if there is an injective sequence of profinite submodules Mi i isin N which is a cofinal sequence for the underlying space of M So countable discrete Λmodules are indprofinite because finitely generateddiscrete Λmodules are finite and so are profinite Λmodules In particular Λwith leftmultiplication is an indprofinite Λmodule Note that since profiniteZmodules are the same as profinite abelian groups indprofinite Zmodules arethe same as indprofinite abelian groupsThen we immediately get the following
Corollary 16 Suppose M is an indprofinite Λmodule with cofinal sequenceMi
(i) Any compact subspace of M is contained in some Mi
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1273
(ii) Closed submodules N of M are indprofinite with cofinal sequence NcapMi
(iii) Quotients of M by closed submodules N are indprofinite with cofinalsequence Mi(N capMi)
(iv) Indprofinite Λmodules are topological Λmodules
As before we call a sequence Mi of profinite submodules making M into anindprofinite Λmodule a cofinal sequence for M
Lemma 17 Indprofinite Λmodules have a fundamental system of neighbourhoods of 0 consisting of open submodules Hence such modules are Hausdorffand totally disconnected
Proof Suppose M has cofinal sequence Mi and suppose U subeM is open with0 isin U by definition U capMi is open in Mi for all i Profinite modules havea fundamental system of neighbourhoods of 0 consisting of open submodulesby [9 Lemma 511] so we can pick an open submodule N0 of M0 such thatN0 sube U capM0 Now we proceed inductively given an open submodule Ni of Mi
such that Ni sube U capMi let f be the quotient map M rarrMNi Then f(U) isopen in MNi by [5 Proposition 13] so f(U)capMi+1Ni is open in Mi+1NiPick an open submodule of Mi+1Ni which is contained in f(U) capMi+1Ni
and write Ni+1 for its preimage in Mi+1 Finally let N be the submodule ofM with cofinal sequence Ni N is open and N sube U as required
Write IP (Λ) for the category whose objects are left indprofinite Λmodulesand whose morphisms M rarr N are Λkmodule homomorphisms We willidentify the category of right indprofinite Λmodules with IP (Λop) in the usualway Given M isin IP (Λ) and a submodule M prime write M prime for the closure of M prime
in M Given MN isin IP (Λ) write HomIPΛ (MN) for the abstract Rmodule
of morphisms M rarr N this makes HomIPΛ (minusminus) into a functor IP (Λ)op times
IP (Λ)rarrMod(R) in the usual way where Mod(R) is the category of abstractRmodules and Rmodule homomorphisms
Proposition 18 IP (Λ) is an additive category with kernels and cokernels
Proof The category is clearly preadditive the biproduct MoplusN is the biproduct of the underlying abstract modules with the topology of M timesk N Theexistence of kernels and cokernels follows from Corollary 16 the cokernel off M rarr N is Nf(M)
Remark 19 The category IP (Λ) is not abelian in general Consider the countable direct sum oplusalefsym0
Z2Z with the discrete topology and the countable direct product
prodalefsym0
Z2Z with the profinite topology Both are indprofinite
Zmodules There is a canonical injective map i oplusZ2Z rarrprod
Z2Z buti(oplusZ2Z) is not closed in
prodZ2Z Moreover oplusZ2Z is not homeomorphic to
i(oplusZ2Z) with the subspace topology because i(oplusZ2Z) is not discrete bythe construction of the product topology
Documenta Mathematica 21 (2016) 1269ndash1312
1274 Marco Boggi and Ged Corob Cook
Given a morphism f M rarr N in a category with kernels and cokernelswe write coim(f) for coker(ker(f)) and im(f) for ker(coker(f)) That iscoim(f) = f(M) with the quotient topology coming from M and im(f) =f(M) with the subspace topology coming from N In an abelian categorycoim(f) = im(f) but the preceding remark shows that this fails in IP (Λ)We say a morphism f M rarr N in IP (Λ) is strict if coim(f) = im(f) Inparticular strict epimorphisms are surjections Note that if M is profinite allmorphisms f M rarr N must be strict because compact subspaces of Hausdorffspaces are closed so that coim(f)rarr im(f) is a continuous bijection of compactHausdorff spaces and hence a topological isomorphism
Proposition 110 Morphisms f M rarr N in IP (Λ) such that f(M) is aclosed subset of N have continuous sections So f is strict in this case and inparticular continuous bijections are isomorphisms
Proof [5 Proposition 16]
Corollary 111 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if f is strict
Proof The decomposition is the usual one M rarr coim(f)gminusrarr im(f) rarr N
for categories with kernels and cokernels Clearly coim(f) = f(M) rarr N isinjective so g is too and hence g is monic Also the settheoretic image ofM rarr im(f) is dense so the settheoretic image of g is too and hence g is epicThen everything follows from Proposition 110
Because IP (Λ) is not abelian it is not obvious what the right notion of exactness is We will say that a chain complex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M Despite the failure of our category to be abelian we can prove the followingSnake Lemma which will be useful later
Lemma 112 Suppose we have a commutative diagram in IP (Λ) of the form
L
f
Mp
g
N
h
0
0 Lprime i M prime N prime
such that the rows are strict exact at MNLprimeM prime and f g h are strict Thenwe have a strict exact sequence
ker(f)rarr ker(g)rarr ker(h)partminusrarr coker(f)rarr coker(g)rarr coker(h)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1275
Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forgetting the topology and working with abstract Λmodules we get the sequencedescribed above from the standard Snake Lemma for abstract modules whichis exact as a sequence of abstract modules This implies that if all the mapsin the sequence are continuous then they have closed settheoretic image andhence the sequence is strict by Proposition 110 To see that part is continuouswe construct it as a composite of continuous maps Since coim(p) = N byProposition 110 again p has a continuous section s1 N rarr M and similarlyi has a continuous section s2 im(i) rarr Lprime Then as usual part = s2gs1 Thecontinuity of the other maps is clear
Proposition 113 The category IP (Λ) has countable colimits
Proof We show first that IP (Λ) has countable direct sums Given a countablecollection Mn n isin N of indprofinite Λmodules write Mni i isin N foreach n for a cofinal sequence for Mn Now consider the injective sequenceNn given by Nn =
prodni=1 Min+1minusi each Nn is a profinite Λmodule so
the sequence defines an indprofinite Λmodule N It is easy to check that theunderlying abstract module of N is
oplusn Mn that each canonical map Mn rarr N
is continuous and that any collection of continuous homomorphisms Mn rarr Pin IP (Λ) induces a continuous N rarr P
Now suppose we have a countable diagram Mn in IP (Λ) Write S for theclosed submodule of
oplusMn generated (topologically) by the elements with jth
component minusx kth component f(x) and all other components 0 for all mapsf Mj rarr Mk in the diagram and all x isin Mj By standard arguments(oplus
Mn)S with the quotient topology is the colimit of the diagram
Remark 114 We get from this construction that given a countable collectionof short strict exact sequences
0rarr Ln rarrMn rarr Nn rarr 0
in IP (Λ) their direct sum
0rarroplus
Ln rarroplus
Mn rarroplus
Nn rarr 0
is strict exact by Proposition 110 because the sequence of underlying modulesis exact So direct sums preserve kernels and cokernels and in particular directsums preserve strict maps because given a countable collection of strict mapsfn in IP (Λ)
coim(oplus
fn) = coker(ker(oplus
fn)) =oplus
coker(ker(fn))
=oplus
ker(coker(fn)) = ker(coker(oplus
fn)) = im(oplus
fn)
Documenta Mathematica 21 (2016) 1269ndash1312
1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

Continuous Cohomology of Profinite Groups 1271
Proposition 11 Given an indprofinite space X defined as the colimit of aninjective sequence Xi of profinite spaces any compact subspace K of X iscontained in some Xi
Proof [5 Proposition 11] proves this under the additional assumption thatthe Xi are profinite groups but the proof does not use this
This shows that compact subspaces of X are exactly the profinite subspacesand that if an indprofinite space X is defined as the colimit of a sequenceXi then the Xi are cofinal in the poset of compact subspaces of X Wecall such a sequence a cofinal sequence for X any cofinal sequence of profinitesubspaces defines X up to homeomorphismA topological space X is called compactly generated if it satisfies the followingcondition a subspace U of X is closed if and only if U cap K is closed in Kfor every compact subspace K of X See [11] for background on such spacesBy the definition of the colimit topology indprofinite spaces are compactlygenerated Indeed a subspace U of an indprofinite space X is closed if andonly if U capXi is closed in Xi for all i if and only if U capK is closed in K forevery compact subspace K of X by Proposition 11
Lemma 12 IPSpace has finite products and coproducts
Proof Given XY isin IPSpace with cofinal sequences Xi Yi we can construct X ⊔ Y using the cofinal sequence Xi ⊔ Yi However it is not clearwhether X times Y with the product topology is indprofinite Instead thanks toProposition 11 the indprofinite space lim
minusrarrXi times Yi is the product of X and Y
it is easy to check that it satisfies the relevant universal property
Moreover by the proposition Xi times Yi is cofinal in the poset of compactsubspaces of X times Y (with the product topology) and hence lim
minusrarrXi times Yi is the
kification of X times Y or in other words it is the product of X and Y in thecategory of compactly generated spaces ndash see [11] for details So we will writeX timesk Y for the product in IPSpaceWe say an abelian group M equipped with an indprofinite topology is an indprofinite abelian group if it satisfies the following condition there is an injectivesequence of profinite subgroups Mi i isin N which is a cofinal sequence for theunderlying space of M It is easy to see that profinite groups and countablediscrete torsion groups are indprofinite Moreover Qp is indprofinite via thecofinal sequence
Zpmiddotpminusrarr Zp
middotpminusrarr middot middot middot (lowast)
Remark 13 It is not obvious that indprofinite abelian groups are topologicalgroups In fact we see below that they are But it is much easier to seethat they are kgroups in the sense of [7] the multiplication map M timesk M =limminusrarrIPSpace
Mi times Mi rarr M is continuous by the definition of colimits The
kgroup intuition will often be more useful
Documenta Mathematica 21 (2016) 1269ndash1312
1272 Marco Boggi and Ged Corob Cook
In the terminology of [5] the indprofinite abelian groups are just the abelianweakly profinite groups We recall some of the basic results of [5]
Proposition 14 Suppose M is an indprofinite abelian group with cofinalsequence Mi
(i) Any compact subspace of M is contained in some Mi
(ii) Closed subgroups N of M are indprofinite with cofinal sequence N capMi
(iii) Quotients of M by closed subgroups N are indprofinite with cofinal sequence Mi(N capMi)
(iv) Indprofinite abelian groups are topological groups
Proof [5 Proposition 11 Proposition 12 Proposition 15]
As before we call a sequence Mi of profinite subgroups making M into anindprofinite group a cofinal sequence for M Suppose from now on that R is a commutative profinite ring and Λ is a profiniteRalgebra
Remark 15 We could define indprofinite rings as colimits of injective sequences (indexed by N) of profinite rings and much of what follows does holdin some sense for such rings but not much is lost by the restriction In particular it would be nice to use the machinery of indprofinite rings to study Qpbut the sequence (lowast) making Qp into an indprofinite abelian group does notmake it into an indprofinite ring because the maps are not maps of rings
We say that M is a left Λkmodule if M is a kgroup equipped with a continuous map ΛtimeskM rarrM A Λkmodule homomorphism M rarr N is a continuousmap which is a homomorphism of the underlying abstract Λmodules BecauseΛ is profinite Λtimesk M = ΛtimesM so ΛtimesM rarrM is continuous Hence if M isa topological group (that is if multiplication M timesM rarrM is continuous) thenit is a topological ΛmoduleWe say that a left Λkmodule M equipped with an indprofinite topology isa left indprofinite Λmodule if there is an injective sequence of profinite submodules Mi i isin N which is a cofinal sequence for the underlying space of M So countable discrete Λmodules are indprofinite because finitely generateddiscrete Λmodules are finite and so are profinite Λmodules In particular Λwith leftmultiplication is an indprofinite Λmodule Note that since profiniteZmodules are the same as profinite abelian groups indprofinite Zmodules arethe same as indprofinite abelian groupsThen we immediately get the following
Corollary 16 Suppose M is an indprofinite Λmodule with cofinal sequenceMi
(i) Any compact subspace of M is contained in some Mi
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1273
(ii) Closed submodules N of M are indprofinite with cofinal sequence NcapMi
(iii) Quotients of M by closed submodules N are indprofinite with cofinalsequence Mi(N capMi)
(iv) Indprofinite Λmodules are topological Λmodules
As before we call a sequence Mi of profinite submodules making M into anindprofinite Λmodule a cofinal sequence for M
Lemma 17 Indprofinite Λmodules have a fundamental system of neighbourhoods of 0 consisting of open submodules Hence such modules are Hausdorffand totally disconnected
Proof Suppose M has cofinal sequence Mi and suppose U subeM is open with0 isin U by definition U capMi is open in Mi for all i Profinite modules havea fundamental system of neighbourhoods of 0 consisting of open submodulesby [9 Lemma 511] so we can pick an open submodule N0 of M0 such thatN0 sube U capM0 Now we proceed inductively given an open submodule Ni of Mi
such that Ni sube U capMi let f be the quotient map M rarrMNi Then f(U) isopen in MNi by [5 Proposition 13] so f(U)capMi+1Ni is open in Mi+1NiPick an open submodule of Mi+1Ni which is contained in f(U) capMi+1Ni
and write Ni+1 for its preimage in Mi+1 Finally let N be the submodule ofM with cofinal sequence Ni N is open and N sube U as required
Write IP (Λ) for the category whose objects are left indprofinite Λmodulesand whose morphisms M rarr N are Λkmodule homomorphisms We willidentify the category of right indprofinite Λmodules with IP (Λop) in the usualway Given M isin IP (Λ) and a submodule M prime write M prime for the closure of M prime
in M Given MN isin IP (Λ) write HomIPΛ (MN) for the abstract Rmodule
of morphisms M rarr N this makes HomIPΛ (minusminus) into a functor IP (Λ)op times
IP (Λ)rarrMod(R) in the usual way where Mod(R) is the category of abstractRmodules and Rmodule homomorphisms
Proposition 18 IP (Λ) is an additive category with kernels and cokernels
Proof The category is clearly preadditive the biproduct MoplusN is the biproduct of the underlying abstract modules with the topology of M timesk N Theexistence of kernels and cokernels follows from Corollary 16 the cokernel off M rarr N is Nf(M)
Remark 19 The category IP (Λ) is not abelian in general Consider the countable direct sum oplusalefsym0
Z2Z with the discrete topology and the countable direct product
prodalefsym0
Z2Z with the profinite topology Both are indprofinite
Zmodules There is a canonical injective map i oplusZ2Z rarrprod
Z2Z buti(oplusZ2Z) is not closed in
prodZ2Z Moreover oplusZ2Z is not homeomorphic to
i(oplusZ2Z) with the subspace topology because i(oplusZ2Z) is not discrete bythe construction of the product topology
Documenta Mathematica 21 (2016) 1269ndash1312
1274 Marco Boggi and Ged Corob Cook
Given a morphism f M rarr N in a category with kernels and cokernelswe write coim(f) for coker(ker(f)) and im(f) for ker(coker(f)) That iscoim(f) = f(M) with the quotient topology coming from M and im(f) =f(M) with the subspace topology coming from N In an abelian categorycoim(f) = im(f) but the preceding remark shows that this fails in IP (Λ)We say a morphism f M rarr N in IP (Λ) is strict if coim(f) = im(f) Inparticular strict epimorphisms are surjections Note that if M is profinite allmorphisms f M rarr N must be strict because compact subspaces of Hausdorffspaces are closed so that coim(f)rarr im(f) is a continuous bijection of compactHausdorff spaces and hence a topological isomorphism
Proposition 110 Morphisms f M rarr N in IP (Λ) such that f(M) is aclosed subset of N have continuous sections So f is strict in this case and inparticular continuous bijections are isomorphisms
Proof [5 Proposition 16]
Corollary 111 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if f is strict
Proof The decomposition is the usual one M rarr coim(f)gminusrarr im(f) rarr N
for categories with kernels and cokernels Clearly coim(f) = f(M) rarr N isinjective so g is too and hence g is monic Also the settheoretic image ofM rarr im(f) is dense so the settheoretic image of g is too and hence g is epicThen everything follows from Proposition 110
Because IP (Λ) is not abelian it is not obvious what the right notion of exactness is We will say that a chain complex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M Despite the failure of our category to be abelian we can prove the followingSnake Lemma which will be useful later
Lemma 112 Suppose we have a commutative diagram in IP (Λ) of the form
L
f
Mp
g
N
h
0
0 Lprime i M prime N prime
such that the rows are strict exact at MNLprimeM prime and f g h are strict Thenwe have a strict exact sequence
ker(f)rarr ker(g)rarr ker(h)partminusrarr coker(f)rarr coker(g)rarr coker(h)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1275
Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forgetting the topology and working with abstract Λmodules we get the sequencedescribed above from the standard Snake Lemma for abstract modules whichis exact as a sequence of abstract modules This implies that if all the mapsin the sequence are continuous then they have closed settheoretic image andhence the sequence is strict by Proposition 110 To see that part is continuouswe construct it as a composite of continuous maps Since coim(p) = N byProposition 110 again p has a continuous section s1 N rarr M and similarlyi has a continuous section s2 im(i) rarr Lprime Then as usual part = s2gs1 Thecontinuity of the other maps is clear
Proposition 113 The category IP (Λ) has countable colimits
Proof We show first that IP (Λ) has countable direct sums Given a countablecollection Mn n isin N of indprofinite Λmodules write Mni i isin N foreach n for a cofinal sequence for Mn Now consider the injective sequenceNn given by Nn =
prodni=1 Min+1minusi each Nn is a profinite Λmodule so
the sequence defines an indprofinite Λmodule N It is easy to check that theunderlying abstract module of N is
oplusn Mn that each canonical map Mn rarr N
is continuous and that any collection of continuous homomorphisms Mn rarr Pin IP (Λ) induces a continuous N rarr P
Now suppose we have a countable diagram Mn in IP (Λ) Write S for theclosed submodule of
oplusMn generated (topologically) by the elements with jth
component minusx kth component f(x) and all other components 0 for all mapsf Mj rarr Mk in the diagram and all x isin Mj By standard arguments(oplus
Mn)S with the quotient topology is the colimit of the diagram
Remark 114 We get from this construction that given a countable collectionof short strict exact sequences
0rarr Ln rarrMn rarr Nn rarr 0
in IP (Λ) their direct sum
0rarroplus
Ln rarroplus
Mn rarroplus
Nn rarr 0
is strict exact by Proposition 110 because the sequence of underlying modulesis exact So direct sums preserve kernels and cokernels and in particular directsums preserve strict maps because given a countable collection of strict mapsfn in IP (Λ)
coim(oplus
fn) = coker(ker(oplus
fn)) =oplus
coker(ker(fn))
=oplus
ker(coker(fn)) = ker(coker(oplus
fn)) = im(oplus
fn)
Documenta Mathematica 21 (2016) 1269ndash1312
1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

1272 Marco Boggi and Ged Corob Cook
In the terminology of [5] the indprofinite abelian groups are just the abelianweakly profinite groups We recall some of the basic results of [5]
Proposition 14 Suppose M is an indprofinite abelian group with cofinalsequence Mi
(i) Any compact subspace of M is contained in some Mi
(ii) Closed subgroups N of M are indprofinite with cofinal sequence N capMi
(iii) Quotients of M by closed subgroups N are indprofinite with cofinal sequence Mi(N capMi)
(iv) Indprofinite abelian groups are topological groups
Proof [5 Proposition 11 Proposition 12 Proposition 15]
As before we call a sequence Mi of profinite subgroups making M into anindprofinite group a cofinal sequence for M Suppose from now on that R is a commutative profinite ring and Λ is a profiniteRalgebra
Remark 15 We could define indprofinite rings as colimits of injective sequences (indexed by N) of profinite rings and much of what follows does holdin some sense for such rings but not much is lost by the restriction In particular it would be nice to use the machinery of indprofinite rings to study Qpbut the sequence (lowast) making Qp into an indprofinite abelian group does notmake it into an indprofinite ring because the maps are not maps of rings
We say that M is a left Λkmodule if M is a kgroup equipped with a continuous map ΛtimeskM rarrM A Λkmodule homomorphism M rarr N is a continuousmap which is a homomorphism of the underlying abstract Λmodules BecauseΛ is profinite Λtimesk M = ΛtimesM so ΛtimesM rarrM is continuous Hence if M isa topological group (that is if multiplication M timesM rarrM is continuous) thenit is a topological ΛmoduleWe say that a left Λkmodule M equipped with an indprofinite topology isa left indprofinite Λmodule if there is an injective sequence of profinite submodules Mi i isin N which is a cofinal sequence for the underlying space of M So countable discrete Λmodules are indprofinite because finitely generateddiscrete Λmodules are finite and so are profinite Λmodules In particular Λwith leftmultiplication is an indprofinite Λmodule Note that since profiniteZmodules are the same as profinite abelian groups indprofinite Zmodules arethe same as indprofinite abelian groupsThen we immediately get the following
Corollary 16 Suppose M is an indprofinite Λmodule with cofinal sequenceMi
(i) Any compact subspace of M is contained in some Mi
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1273
(ii) Closed submodules N of M are indprofinite with cofinal sequence NcapMi
(iii) Quotients of M by closed submodules N are indprofinite with cofinalsequence Mi(N capMi)
(iv) Indprofinite Λmodules are topological Λmodules
As before we call a sequence Mi of profinite submodules making M into anindprofinite Λmodule a cofinal sequence for M
Lemma 17 Indprofinite Λmodules have a fundamental system of neighbourhoods of 0 consisting of open submodules Hence such modules are Hausdorffand totally disconnected
Proof Suppose M has cofinal sequence Mi and suppose U subeM is open with0 isin U by definition U capMi is open in Mi for all i Profinite modules havea fundamental system of neighbourhoods of 0 consisting of open submodulesby [9 Lemma 511] so we can pick an open submodule N0 of M0 such thatN0 sube U capM0 Now we proceed inductively given an open submodule Ni of Mi
such that Ni sube U capMi let f be the quotient map M rarrMNi Then f(U) isopen in MNi by [5 Proposition 13] so f(U)capMi+1Ni is open in Mi+1NiPick an open submodule of Mi+1Ni which is contained in f(U) capMi+1Ni
and write Ni+1 for its preimage in Mi+1 Finally let N be the submodule ofM with cofinal sequence Ni N is open and N sube U as required
Write IP (Λ) for the category whose objects are left indprofinite Λmodulesand whose morphisms M rarr N are Λkmodule homomorphisms We willidentify the category of right indprofinite Λmodules with IP (Λop) in the usualway Given M isin IP (Λ) and a submodule M prime write M prime for the closure of M prime
in M Given MN isin IP (Λ) write HomIPΛ (MN) for the abstract Rmodule
of morphisms M rarr N this makes HomIPΛ (minusminus) into a functor IP (Λ)op times
IP (Λ)rarrMod(R) in the usual way where Mod(R) is the category of abstractRmodules and Rmodule homomorphisms
Proposition 18 IP (Λ) is an additive category with kernels and cokernels
Proof The category is clearly preadditive the biproduct MoplusN is the biproduct of the underlying abstract modules with the topology of M timesk N Theexistence of kernels and cokernels follows from Corollary 16 the cokernel off M rarr N is Nf(M)
Remark 19 The category IP (Λ) is not abelian in general Consider the countable direct sum oplusalefsym0
Z2Z with the discrete topology and the countable direct product
prodalefsym0
Z2Z with the profinite topology Both are indprofinite
Zmodules There is a canonical injective map i oplusZ2Z rarrprod
Z2Z buti(oplusZ2Z) is not closed in
prodZ2Z Moreover oplusZ2Z is not homeomorphic to
i(oplusZ2Z) with the subspace topology because i(oplusZ2Z) is not discrete bythe construction of the product topology
Documenta Mathematica 21 (2016) 1269ndash1312
1274 Marco Boggi and Ged Corob Cook
Given a morphism f M rarr N in a category with kernels and cokernelswe write coim(f) for coker(ker(f)) and im(f) for ker(coker(f)) That iscoim(f) = f(M) with the quotient topology coming from M and im(f) =f(M) with the subspace topology coming from N In an abelian categorycoim(f) = im(f) but the preceding remark shows that this fails in IP (Λ)We say a morphism f M rarr N in IP (Λ) is strict if coim(f) = im(f) Inparticular strict epimorphisms are surjections Note that if M is profinite allmorphisms f M rarr N must be strict because compact subspaces of Hausdorffspaces are closed so that coim(f)rarr im(f) is a continuous bijection of compactHausdorff spaces and hence a topological isomorphism
Proposition 110 Morphisms f M rarr N in IP (Λ) such that f(M) is aclosed subset of N have continuous sections So f is strict in this case and inparticular continuous bijections are isomorphisms
Proof [5 Proposition 16]
Corollary 111 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if f is strict
Proof The decomposition is the usual one M rarr coim(f)gminusrarr im(f) rarr N
for categories with kernels and cokernels Clearly coim(f) = f(M) rarr N isinjective so g is too and hence g is monic Also the settheoretic image ofM rarr im(f) is dense so the settheoretic image of g is too and hence g is epicThen everything follows from Proposition 110
Because IP (Λ) is not abelian it is not obvious what the right notion of exactness is We will say that a chain complex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M Despite the failure of our category to be abelian we can prove the followingSnake Lemma which will be useful later
Lemma 112 Suppose we have a commutative diagram in IP (Λ) of the form
L
f
Mp
g
N
h
0
0 Lprime i M prime N prime
such that the rows are strict exact at MNLprimeM prime and f g h are strict Thenwe have a strict exact sequence
ker(f)rarr ker(g)rarr ker(h)partminusrarr coker(f)rarr coker(g)rarr coker(h)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1275
Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forgetting the topology and working with abstract Λmodules we get the sequencedescribed above from the standard Snake Lemma for abstract modules whichis exact as a sequence of abstract modules This implies that if all the mapsin the sequence are continuous then they have closed settheoretic image andhence the sequence is strict by Proposition 110 To see that part is continuouswe construct it as a composite of continuous maps Since coim(p) = N byProposition 110 again p has a continuous section s1 N rarr M and similarlyi has a continuous section s2 im(i) rarr Lprime Then as usual part = s2gs1 Thecontinuity of the other maps is clear
Proposition 113 The category IP (Λ) has countable colimits
Proof We show first that IP (Λ) has countable direct sums Given a countablecollection Mn n isin N of indprofinite Λmodules write Mni i isin N foreach n for a cofinal sequence for Mn Now consider the injective sequenceNn given by Nn =
prodni=1 Min+1minusi each Nn is a profinite Λmodule so
the sequence defines an indprofinite Λmodule N It is easy to check that theunderlying abstract module of N is
oplusn Mn that each canonical map Mn rarr N
is continuous and that any collection of continuous homomorphisms Mn rarr Pin IP (Λ) induces a continuous N rarr P
Now suppose we have a countable diagram Mn in IP (Λ) Write S for theclosed submodule of
oplusMn generated (topologically) by the elements with jth
component minusx kth component f(x) and all other components 0 for all mapsf Mj rarr Mk in the diagram and all x isin Mj By standard arguments(oplus
Mn)S with the quotient topology is the colimit of the diagram
Remark 114 We get from this construction that given a countable collectionof short strict exact sequences
0rarr Ln rarrMn rarr Nn rarr 0
in IP (Λ) their direct sum
0rarroplus
Ln rarroplus
Mn rarroplus
Nn rarr 0
is strict exact by Proposition 110 because the sequence of underlying modulesis exact So direct sums preserve kernels and cokernels and in particular directsums preserve strict maps because given a countable collection of strict mapsfn in IP (Λ)
coim(oplus
fn) = coker(ker(oplus
fn)) =oplus
coker(ker(fn))
=oplus
ker(coker(fn)) = ker(coker(oplus
fn)) = im(oplus
fn)
Documenta Mathematica 21 (2016) 1269ndash1312
1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

Continuous Cohomology of Profinite Groups 1273
(ii) Closed submodules N of M are indprofinite with cofinal sequence NcapMi
(iii) Quotients of M by closed submodules N are indprofinite with cofinalsequence Mi(N capMi)
(iv) Indprofinite Λmodules are topological Λmodules
As before we call a sequence Mi of profinite submodules making M into anindprofinite Λmodule a cofinal sequence for M
Lemma 17 Indprofinite Λmodules have a fundamental system of neighbourhoods of 0 consisting of open submodules Hence such modules are Hausdorffand totally disconnected
Proof Suppose M has cofinal sequence Mi and suppose U subeM is open with0 isin U by definition U capMi is open in Mi for all i Profinite modules havea fundamental system of neighbourhoods of 0 consisting of open submodulesby [9 Lemma 511] so we can pick an open submodule N0 of M0 such thatN0 sube U capM0 Now we proceed inductively given an open submodule Ni of Mi
such that Ni sube U capMi let f be the quotient map M rarrMNi Then f(U) isopen in MNi by [5 Proposition 13] so f(U)capMi+1Ni is open in Mi+1NiPick an open submodule of Mi+1Ni which is contained in f(U) capMi+1Ni
and write Ni+1 for its preimage in Mi+1 Finally let N be the submodule ofM with cofinal sequence Ni N is open and N sube U as required
Write IP (Λ) for the category whose objects are left indprofinite Λmodulesand whose morphisms M rarr N are Λkmodule homomorphisms We willidentify the category of right indprofinite Λmodules with IP (Λop) in the usualway Given M isin IP (Λ) and a submodule M prime write M prime for the closure of M prime
in M Given MN isin IP (Λ) write HomIPΛ (MN) for the abstract Rmodule
of morphisms M rarr N this makes HomIPΛ (minusminus) into a functor IP (Λ)op times
IP (Λ)rarrMod(R) in the usual way where Mod(R) is the category of abstractRmodules and Rmodule homomorphisms
Proposition 18 IP (Λ) is an additive category with kernels and cokernels
Proof The category is clearly preadditive the biproduct MoplusN is the biproduct of the underlying abstract modules with the topology of M timesk N Theexistence of kernels and cokernels follows from Corollary 16 the cokernel off M rarr N is Nf(M)
Remark 19 The category IP (Λ) is not abelian in general Consider the countable direct sum oplusalefsym0
Z2Z with the discrete topology and the countable direct product
prodalefsym0
Z2Z with the profinite topology Both are indprofinite
Zmodules There is a canonical injective map i oplusZ2Z rarrprod
Z2Z buti(oplusZ2Z) is not closed in
prodZ2Z Moreover oplusZ2Z is not homeomorphic to
i(oplusZ2Z) with the subspace topology because i(oplusZ2Z) is not discrete bythe construction of the product topology
Documenta Mathematica 21 (2016) 1269ndash1312
1274 Marco Boggi and Ged Corob Cook
Given a morphism f M rarr N in a category with kernels and cokernelswe write coim(f) for coker(ker(f)) and im(f) for ker(coker(f)) That iscoim(f) = f(M) with the quotient topology coming from M and im(f) =f(M) with the subspace topology coming from N In an abelian categorycoim(f) = im(f) but the preceding remark shows that this fails in IP (Λ)We say a morphism f M rarr N in IP (Λ) is strict if coim(f) = im(f) Inparticular strict epimorphisms are surjections Note that if M is profinite allmorphisms f M rarr N must be strict because compact subspaces of Hausdorffspaces are closed so that coim(f)rarr im(f) is a continuous bijection of compactHausdorff spaces and hence a topological isomorphism
Proposition 110 Morphisms f M rarr N in IP (Λ) such that f(M) is aclosed subset of N have continuous sections So f is strict in this case and inparticular continuous bijections are isomorphisms
Proof [5 Proposition 16]
Corollary 111 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if f is strict
Proof The decomposition is the usual one M rarr coim(f)gminusrarr im(f) rarr N
for categories with kernels and cokernels Clearly coim(f) = f(M) rarr N isinjective so g is too and hence g is monic Also the settheoretic image ofM rarr im(f) is dense so the settheoretic image of g is too and hence g is epicThen everything follows from Proposition 110
Because IP (Λ) is not abelian it is not obvious what the right notion of exactness is We will say that a chain complex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M Despite the failure of our category to be abelian we can prove the followingSnake Lemma which will be useful later
Lemma 112 Suppose we have a commutative diagram in IP (Λ) of the form
L
f
Mp
g
N
h
0
0 Lprime i M prime N prime
such that the rows are strict exact at MNLprimeM prime and f g h are strict Thenwe have a strict exact sequence
ker(f)rarr ker(g)rarr ker(h)partminusrarr coker(f)rarr coker(g)rarr coker(h)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1275
Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forgetting the topology and working with abstract Λmodules we get the sequencedescribed above from the standard Snake Lemma for abstract modules whichis exact as a sequence of abstract modules This implies that if all the mapsin the sequence are continuous then they have closed settheoretic image andhence the sequence is strict by Proposition 110 To see that part is continuouswe construct it as a composite of continuous maps Since coim(p) = N byProposition 110 again p has a continuous section s1 N rarr M and similarlyi has a continuous section s2 im(i) rarr Lprime Then as usual part = s2gs1 Thecontinuity of the other maps is clear
Proposition 113 The category IP (Λ) has countable colimits
Proof We show first that IP (Λ) has countable direct sums Given a countablecollection Mn n isin N of indprofinite Λmodules write Mni i isin N foreach n for a cofinal sequence for Mn Now consider the injective sequenceNn given by Nn =
prodni=1 Min+1minusi each Nn is a profinite Λmodule so
the sequence defines an indprofinite Λmodule N It is easy to check that theunderlying abstract module of N is
oplusn Mn that each canonical map Mn rarr N
is continuous and that any collection of continuous homomorphisms Mn rarr Pin IP (Λ) induces a continuous N rarr P
Now suppose we have a countable diagram Mn in IP (Λ) Write S for theclosed submodule of
oplusMn generated (topologically) by the elements with jth
component minusx kth component f(x) and all other components 0 for all mapsf Mj rarr Mk in the diagram and all x isin Mj By standard arguments(oplus
Mn)S with the quotient topology is the colimit of the diagram
Remark 114 We get from this construction that given a countable collectionof short strict exact sequences
0rarr Ln rarrMn rarr Nn rarr 0
in IP (Λ) their direct sum
0rarroplus
Ln rarroplus
Mn rarroplus
Nn rarr 0
is strict exact by Proposition 110 because the sequence of underlying modulesis exact So direct sums preserve kernels and cokernels and in particular directsums preserve strict maps because given a countable collection of strict mapsfn in IP (Λ)
coim(oplus
fn) = coker(ker(oplus
fn)) =oplus
coker(ker(fn))
=oplus
ker(coker(fn)) = ker(coker(oplus
fn)) = im(oplus
fn)
Documenta Mathematica 21 (2016) 1269ndash1312
1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

1274 Marco Boggi and Ged Corob Cook
Given a morphism f M rarr N in a category with kernels and cokernelswe write coim(f) for coker(ker(f)) and im(f) for ker(coker(f)) That iscoim(f) = f(M) with the quotient topology coming from M and im(f) =f(M) with the subspace topology coming from N In an abelian categorycoim(f) = im(f) but the preceding remark shows that this fails in IP (Λ)We say a morphism f M rarr N in IP (Λ) is strict if coim(f) = im(f) Inparticular strict epimorphisms are surjections Note that if M is profinite allmorphisms f M rarr N must be strict because compact subspaces of Hausdorffspaces are closed so that coim(f)rarr im(f) is a continuous bijection of compactHausdorff spaces and hence a topological isomorphism
Proposition 110 Morphisms f M rarr N in IP (Λ) such that f(M) is aclosed subset of N have continuous sections So f is strict in this case and inparticular continuous bijections are isomorphisms
Proof [5 Proposition 16]
Corollary 111 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if f is strict
Proof The decomposition is the usual one M rarr coim(f)gminusrarr im(f) rarr N
for categories with kernels and cokernels Clearly coim(f) = f(M) rarr N isinjective so g is too and hence g is monic Also the settheoretic image ofM rarr im(f) is dense so the settheoretic image of g is too and hence g is epicThen everything follows from Proposition 110
Because IP (Λ) is not abelian it is not obvious what the right notion of exactness is We will say that a chain complex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M Despite the failure of our category to be abelian we can prove the followingSnake Lemma which will be useful later
Lemma 112 Suppose we have a commutative diagram in IP (Λ) of the form
L
f
Mp
g
N
h
0
0 Lprime i M prime N prime
such that the rows are strict exact at MNLprimeM prime and f g h are strict Thenwe have a strict exact sequence
ker(f)rarr ker(g)rarr ker(h)partminusrarr coker(f)rarr coker(g)rarr coker(h)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1275
Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forgetting the topology and working with abstract Λmodules we get the sequencedescribed above from the standard Snake Lemma for abstract modules whichis exact as a sequence of abstract modules This implies that if all the mapsin the sequence are continuous then they have closed settheoretic image andhence the sequence is strict by Proposition 110 To see that part is continuouswe construct it as a composite of continuous maps Since coim(p) = N byProposition 110 again p has a continuous section s1 N rarr M and similarlyi has a continuous section s2 im(i) rarr Lprime Then as usual part = s2gs1 Thecontinuity of the other maps is clear
Proposition 113 The category IP (Λ) has countable colimits
Proof We show first that IP (Λ) has countable direct sums Given a countablecollection Mn n isin N of indprofinite Λmodules write Mni i isin N foreach n for a cofinal sequence for Mn Now consider the injective sequenceNn given by Nn =
prodni=1 Min+1minusi each Nn is a profinite Λmodule so
the sequence defines an indprofinite Λmodule N It is easy to check that theunderlying abstract module of N is
oplusn Mn that each canonical map Mn rarr N
is continuous and that any collection of continuous homomorphisms Mn rarr Pin IP (Λ) induces a continuous N rarr P
Now suppose we have a countable diagram Mn in IP (Λ) Write S for theclosed submodule of
oplusMn generated (topologically) by the elements with jth
component minusx kth component f(x) and all other components 0 for all mapsf Mj rarr Mk in the diagram and all x isin Mj By standard arguments(oplus
Mn)S with the quotient topology is the colimit of the diagram
Remark 114 We get from this construction that given a countable collectionof short strict exact sequences
0rarr Ln rarrMn rarr Nn rarr 0
in IP (Λ) their direct sum
0rarroplus
Ln rarroplus
Mn rarroplus
Nn rarr 0
is strict exact by Proposition 110 because the sequence of underlying modulesis exact So direct sums preserve kernels and cokernels and in particular directsums preserve strict maps because given a countable collection of strict mapsfn in IP (Λ)
coim(oplus
fn) = coker(ker(oplus
fn)) =oplus
coker(ker(fn))
=oplus
ker(coker(fn)) = ker(coker(oplus
fn)) = im(oplus
fn)
Documenta Mathematica 21 (2016) 1269ndash1312
1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

Continuous Cohomology of Profinite Groups 1275
Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forgetting the topology and working with abstract Λmodules we get the sequencedescribed above from the standard Snake Lemma for abstract modules whichis exact as a sequence of abstract modules This implies that if all the mapsin the sequence are continuous then they have closed settheoretic image andhence the sequence is strict by Proposition 110 To see that part is continuouswe construct it as a composite of continuous maps Since coim(p) = N byProposition 110 again p has a continuous section s1 N rarr M and similarlyi has a continuous section s2 im(i) rarr Lprime Then as usual part = s2gs1 Thecontinuity of the other maps is clear
Proposition 113 The category IP (Λ) has countable colimits
Proof We show first that IP (Λ) has countable direct sums Given a countablecollection Mn n isin N of indprofinite Λmodules write Mni i isin N foreach n for a cofinal sequence for Mn Now consider the injective sequenceNn given by Nn =
prodni=1 Min+1minusi each Nn is a profinite Λmodule so
the sequence defines an indprofinite Λmodule N It is easy to check that theunderlying abstract module of N is
oplusn Mn that each canonical map Mn rarr N
is continuous and that any collection of continuous homomorphisms Mn rarr Pin IP (Λ) induces a continuous N rarr P
Now suppose we have a countable diagram Mn in IP (Λ) Write S for theclosed submodule of
oplusMn generated (topologically) by the elements with jth
component minusx kth component f(x) and all other components 0 for all mapsf Mj rarr Mk in the diagram and all x isin Mj By standard arguments(oplus
Mn)S with the quotient topology is the colimit of the diagram
Remark 114 We get from this construction that given a countable collectionof short strict exact sequences
0rarr Ln rarrMn rarr Nn rarr 0
in IP (Λ) their direct sum
0rarroplus
Ln rarroplus
Mn rarroplus
Nn rarr 0
is strict exact by Proposition 110 because the sequence of underlying modulesis exact So direct sums preserve kernels and cokernels and in particular directsums preserve strict maps because given a countable collection of strict mapsfn in IP (Λ)
coim(oplus
fn) = coker(ker(oplus
fn)) =oplus
coker(ker(fn))
=oplus
ker(coker(fn)) = ker(coker(oplus
fn)) = im(oplus
fn)
Documenta Mathematica 21 (2016) 1269ndash1312
1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

1276 Marco Boggi and Ged Corob Cook
Lemma 115 (i) For MN isin IP (Λ) let Mi Nj cofinal sequences of M
and N respectively HomIPΛ (MN) = lim
larrminusilimminusrarrj
HomIPΛ (Mi Nj) in the
category of Rmodules
(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the Rmodule of continuousmaps X rarr N Then C(XN) = lim
larrminusilimminusrarrj
C(Xi Nj)
Proof (i) Since M = limminusrarrIP (Λ)
Mi we have that
HomIPΛ (MN) = lim
larrminusHomIP
Λ (Mi N)
Since the Nj are cofinal for N every continuous map Mi rarr N factors
through some Nj so HomIPΛ (Mi N) = lim
minusrarrHomIP
Λ (Mi Nj)
(ii) Similarly
Given X isin IPSpace as before define a module FX isin IP (Λ) in the followingway let FXi be the free profinite Λmodule on Xi The maps Xi rarr Xi+1
induce maps FXi rarr FXi+1 of profinite Λmodules and hence we get an indprofinite Λmodule with cofinal sequence FXi Write FX for this modulewhich we will call the free indprofinite Λmodule on X
Proposition 116 Suppose X isin IPSpace and N isin IP (Λ) Then we haveHomIP
Λ (FXN) = C(XN) naturally in X and N
Proof First recall that by the definition of free profinite modules there holdsHomIP
Λ (FXN) = C(XN) when X and N are profinite Then by Lemma115
HomIPΛ (FXN) = lim
larrminusi
limminusrarrj
HomIPΛ (FXi Nj) = lim
larrminusi
limminusrarrj
C(Xi Nj) = C(XN)
The isomorphism is natural because HomIPΛ (Fminusminus) and C(minusminus) are both
bifunctors
We call P isin IP (Λ) projective if
0rarr HomIPΛ (PL)rarr HomIP
Λ (PM)rarr HomIPΛ (PN)rarr 0
is an exact sequence in Mod(R) whenever
0rarr LrarrM rarr N rarr 0
is strict exact We will say IP (Λ) has enough projectives if for everyM isin IP (Λ)there is a projective P and a strict epimorphism P rarrM
Corollary 117 IP (Λ) has enough projectives
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

Continuous Cohomology of Profinite Groups 1277
Proof By Proposition 116 and Proposition 110 FX is projective for all X isinIPSpace So given M isin IP (Λ) FM has the required property the identityM rarr M induces a canonical lsquoevaluation maprsquo ε FM rarr M which is strictepic because it is a surjection
Lemma 118 Projective modules in IP (Λ) are summands of free ones
Proof Given a projective P isin IP (Λ) pick a free module F and a strict epi
morphism f F rarr P By definition the map HomIPΛ (P F )
flowast
minusrarr HomIPΛ (P P )
induced by f is a surjection so there is some morphism g P rarr F such thatflowast(g) = gf = idP Then we get that the map ker(f)oplus P rarr F is a continuousbijection and hence an isomorphism by Proposition 110
Remarks 119 (i) We can also define the class of strictly free modules to befree indprofinite modules on indprofinite spaces X which have the formof a disjoint union of profinite spaces Xi By the universal properties ofcoproducts and free modules we immediately get FX =
oplusFXi More
over for every indprofinite space Y there is some X of this form witha surjection X rarr Y given a cofinal sequence Yi in Y let X =
⊔Yi
and the identity maps Yi rarr Yi induce the required map X rarr Y Thenthe same argument as before shows that projective modules in IP (Λ) aresummands of strictly free ones
(ii) Note that a profinite module in IP (Λ) is projective in IP (Λ) if and only ifit is projective in the category of profinite Λmodules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows
2 ProDiscrete Modules
Write PD(Λ) for the category of left prodiscrete Λmodules the objects M inthis category are countable inverse limits as topological Λmodules of discreteΛmodules M i i isin N the morphisms are continuous Λmodule homomorphisms So discrete torsion Λmodules are prodiscrete and so are secondcountable profinite Λmodules by [9 Proposition 261 Lemma 511] and inparticular Λ with leftmultiplication is a prodiscrete Λmodule if Λ is secondcountable Moreover Qp is a prodiscrete Zmodule via the sequence
middot middot middotmiddotpminusrarr QpZp
middotpminusrarr QpZp
We will identify the category of right prodiscrete Λmodules with PD(Λop) inthe usual way
Lemma 21 Prodiscrete Λmodules are firstcountable
Proof We can construct M = limlarrminus
M i as a closed subspace ofprod
M i Each M i
is firstcountable because it is discrete and firstcountability is closed undercountable products and subspaces
Documenta Mathematica 21 (2016) 1269ndash1312
1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

1278 Marco Boggi and Ged Corob Cook
Remarks 22 (i) This shows that Λ itself can be regarded as a prodiscreteΛmodule if and only if it is firstcountable if and only if it is secondcountable by [9 Proposition 261] Rings of interest are often second
countable this class includes for example Zp Z Qp and the completedgroup ring RJGK when R and G are secondcountable
(ii) Since firstcountable spaces are always compactly generated by [11Proposition 16] prodiscrete Λmodules are compactly generated as topological spaces In fact more is true Given a prodiscrete Λmodule Mwhich is the inverse limit of a countable sequence M i of finite quotientssuppose X is a compact subspace of M and write X i for the image of Xin M i By compactness each X i is finite Let N i be the submodule ofM i generated by X i because X i is finite Λ is compact and M i is discrete torsion N i is finite Hence N = lim
larrminusN i is a profinite Λsubmodule
of M containing X So prodiscrete modules M are compactly generatedby their profinite submodules N in the sense that a subspace U of M isclosed if and only if U capN is closed in N for all N
Lemma 23 Prodiscrete Λmodules are metrisable and complete
Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]
In general prodiscrete Λmodules need not be secondcountable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result
Lemma 24 Suppose a Λmodule M has a topology which makes it prodiscreteand indprofinite (as a Λmodule) Then M is secondcountable and locallycompact
Proof As an indprofinite Λmodule take a cofinal sequence of profinite submodules Mi For any discrete quotient N of M the image of each Mi in N iscompact and hence finite and N is the union of these images so N is countable Then ifM is the inverse limit of a countable sequence of discrete quotientsM j each M j is countable and M can be identified with a closed subspace ofprod
M j so M is secondcountable because secondcountability is closed undercountable products and subspaces By Proposition 23 M is a Baire spaceand hence by the Baire category theorem one of the Mi must be open Theresult follows
Proposition 25 Suppose M is a prodiscrete Λmodule which is the inverselimit of a sequence of discrete quotient modules M i Let U i = ker(M rarrM i)
(i) The sequence M i is cofinal in the poset of all discrete quotient modulesof M
(ii) A closed submodule N of M is prodiscrete with a cofinal sequenceN(N cap U i)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory of discrete Λmodules Then I is injective in PD(Λ)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1281
Proof We know HomPDΛ (minus I) is exact on discrete Λmodules Remark 29
shows that we can write short strict exact sequences of prodiscrete Λmodulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD
Λ (minus I) gives a direct system of short exact sequences of Rmodules the exactness of such direct limitsis wellknown
In particular we get that QZ with the discrete topology is injective in PD(Z)
ndash it is injective among discrete Zmodules (ie torsion abelian groups) by Baerrsquoslemma because it is divisible (see [14 231])Given M isin IP (Λ) with a cofinal sequence Mi and N isin PD(Λ) with acofinal sequence N j we can consider the continuous group homomorphismsf M rarr N which are compatible with the Λaction ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λmodules and continuous Λmodule homomorphisms We can consider IP (Λ)and PD(Λ) as full subcategories of T (Λ) and observe that M = lim
minusrarrT (Λ)Mi
and N = limlarrminusT (Λ)
N j We write HomTΛ(MN) for the Rmodule of morphisms
M rarr N in T (Λ) For the following lemma this will denote an abstract Rmodule after which we will define a topology on HomT
Λ(MN) making it intoa topological Rmodule
Lemma 212 As abstract Rmodules HomTΛ(MN) = lim
larrminusijHomT
Λ(Mi Nj)
We may give each HomTΛ(Mi N
j) the discrete topology which is also thecompactopen topology in this case Then we make lim
larrminusHomT
Λ(Mi Nj) into
a topological Rmodule by giving it the limit topology giving HomTΛ(MN)
this topology therefore makes it into a prodiscrete Rmodule From now onHomT
Λ(MN) will be understood to have this topology The topology thusconstructed is welldefined because the Mi are cofinal for M and the N j
cofinal for N Moreover given a morphism M rarr M prime in IP (Λ) this construction makes the induced map HomT
Λ(Mprime N) rarr HomT
Λ(MN) continuousand similarly in the second variable so that HomT
Λ(minusminus) becomes a functorIP (Λ)op times PD(Λ) rarr PD(R) Of course the case when M and N are rightΛmodules behaves in the same way we may express this by treating MN asleft Λopmodules and writing HomT
Λop(MN) in this caseMore generally given a chain complex
middot middot middotd1minusrarrM1
d0minusrarrM0dminus1
minusminusrarr middot middot middot
in IP (Λ) and a cochain complex
middot middot middotdminus1
minusminusrarr N0 d0
minusrarr N1 d1
minusrarr middot middot middot
in PD(Λ) both bounded below let us define the double cochain complexHomT
Λ(Mp Nq) with the obvious horizontal maps and with the vertical maps
Documenta Mathematica 21 (2016) 1269ndash1312
1282 Marco Boggi and Ged Corob Cook
defined in the obvious way except that they are multiplied by minus1 whenever p isodd this makes Tot(HomT
Λ(Mp Nq)) into a cochain complex which we denote
by HomTΛ(MN) Each term in the total complex is the sum of finitely many
prodiscrete Rmodules becauseM andN are bounded below so HomTΛ(MN)
is a complex in PD(R)
Suppose ΘΦ are profinite Ralgebras Then let PD(ΘminusΦ) be the category ofprodiscrete ΘminusΦbimodules and continuous ΘminusΦhomomorphisms If M isan indprofinite Λ minusΘbimodule and N is a prodiscrete Λminus Φbimodule onecan make HomT
Λ(MN) into a prodiscrete ΘminusΦbimodule in the same way asin the abstract case We leave the details to the reader
3 Pontryagin Duality
Lemma 31 Suppose that I is a discrete Λmodule which is injective in PD(Λ)Then HomT
Λ(minus I) sends short strict exact sequences of indprofinite Λmodulesto short strict exact sequences of prodiscrete Rmodules
Proof Proposition 110 shows that we can write short strict exact sequencesof indprofinite Λmodules as injective direct limits of short exact sequences ofprofinite modules in IP (Λ) and then [9 Exercise 547(b)] shows that applyingHomT
Λ(minus I) gives a surjective inverse system of short exact sequences of discreteRmodules the inverse limit of these is strict exact by Remark 29
In particular this applies when I = QZ with the discrete topology as a
Zmodule
Consider QZ with the discrete topology as an indprofinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anindprofinite abelian group by forgetting the Λaction then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT
Z(minusQZ)
to get a prodiscrete abelian group We can endow each HomTZ(MiQZ) with
the structure of a right Λmodule such that the Λaction is continuous by[9 p165] Taking inverse limits we can therefore make HomT
Z(MQZ) into
a prodiscrete right Λmodule which we denote by Mlowast As before lowast gives acontravariant functor IP (Λ) rarr PD(Λop) Lemma 31 now has the followingimmediate consequence
Corollary 32 The functor lowast IP (Λ) rarr PD(Λop) maps short strict exactsequences to short strict exact sequences
Suppose instead that M isin PD(Λ) with a cofinal sequence M i As beforewe can think of M as a prodiscrete abelian group by forgetting the Λactionand then M i is a cofinal sequence of discrete abelian groups Recall that
as (abstract) Zmodules HomPDZ
(MQZ) sim= limminusrarri
HomPDZ
(M iQZ) We can
endow each HomPDZ
(M iQZ) with the structure of a profinite right Λmodule
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1283
by [9 p165] Taking direct limits we then make HomPDZ
(MQZ) into an indprofinite right Λmodule which we denote byMlowast and in the same way as before
lowast gives a functor PD(Λ)rarr IP (Λop)Note that lowast also maps short strict exact sequences to short strict exact sequences by Lemma 211 and Proposition 110 Note too that both lowast and lowast
send profinite modules to discrete modules and vice versa on such modulesthey give the same result as the usual Pontryagin duality functor of [9 Section29]
Theorem 33 (Pontryagin duality) The composite functors IP (Λ)minuslowast
minusminusrarr
PD(Λop)minuslowastminusminusrarr IP (Λ) and PD(Λ)
minuslowastminusminusrarr IP (Λop)minuslowast
minusminusrarr PD(Λ) are naturally isomorphic to the identity so that IP (Λ) and PD(Λ) are dually equivalent
Proof We give a proof for lowast lowast the proof for lowast lowast is similar Given M isin
IP (Λ) with a cofinal sequence Mi by construction (Mlowast)lowast has cofinal sequence(Mlowast
i )lowast By [9 p165] the functors lowast and lowast give a dual equivalence between thecategories of profinite and discrete Λmodules so we have natural isomorphismsMi rarr (Mlowast
i )lowast for each i and the result follows
From now on by abuse of notation we will follow convention by writing lowast forboth the functors lowast and lowast
Corollary 34 Pontryagin duality preserves the canonical decomposition ofmorphisms More precisely given a morphism f M rarr N in IP (Λ) im(f)lowast =coim(flowast) and im(flowast) = coim(f)lowast In particular flowast is strict if and only if f isSimilarly for morphisms in PD(Λ)
Proof This follows from Pontryagin duality and the duality between the definitions of im and coim For the final observation note that by Corollary 111and Corollary 28
flowast is stricthArr im(flowast) = coim(flowast)
hArr im(f) = coim(f)
hArr f is strict
Corollary 35 (i) PD(Λ) has countable limits
(ii) Direct products in PD(Λ) preserve kernels and cokernels and hence strictmaps
(iii) PD(Λ) has enough injectives for every M isin PD(Λ) there is an injective I and a strict monomorphism M rarr I A discrete Λmodule I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛmodules
Documenta Mathematica 21 (2016) 1269ndash1312
1284 Marco Boggi and Ged Corob Cook
(iv) Every injective in PD(Λ) is a summand of a strictly cofree one ie onewhose Pontryagin dual is strictly free
(v) Countable products of strict exact sequences in PD(Λ) are strict exact
(vi) Let P be a profinite Λmodule which is projective in IP (Λ) Then the functor HomT
Λ(Pminus) sends strict exact sequences of prodiscrete Λmodules tostrict exact sequences of prodiscrete Rmodules
Example 36 It is easy to check that Zlowast = QZ and Zlowastp = QpZp Then
Qlowastp = (lim
minusrarr(Zp
middotpminusrarr Zp
middotpminusrarr middot middot middot ))lowast = lim
larrminus(middot middot middot
middotpminusrarr QpZp
middotpminusrarr QpZp) = Qp
The topology defined on Mlowast = HomTZ(MQZ) when M is an indprofinite
Λmodule coincides with the compactopen topology because the (discrete)topology on each HomT
Z(MiQZ) is the compactopen topology and every
compact subspace of M is contained in some Mi by Proposition 11 Similarly for a prodiscrete Λmodule N every compact subspace of N is contained in some profinite submodule L by Remark 22(ii) and so the compactopen topology on HomPD
Z(NQZ) coincides with the limit topology on
limlarrminusT (Λ)
HomPDZ
(LQZ) where the limit is taken over all profinite submodules
of N and each HomPDZ
(LQZ) is given the (discrete) compactopen topology
Proposition 37 The compactopen topology on HomPDZ
(NQZ) coincideswith the topology defined on Nlowast
Proof By the preceding remarks HomPDZ
(NQZ) with the compactopentopology is just lim
larrminusprofinite LleNLlowast So the canonical map Nlowast rarr lim
larrminusLlowast is a
continuous bijection we need to check it is open By Lemma 17 it suffices tocheck this for open submodules K of Nlowast Because K is open NlowastK is discrete so (NlowastK)lowast is a profinite submodule of N Therefore there is a canonicalcontinuous map lim
larrminusLlowast rarr (NlowastK)lowastlowast = NlowastK whose kernel is open because
NlowastK is discrete This kernel is K and the result follows
Corollary 38 The topology on indprofinite Λmodules is complete Hausdorff and totally disconnected
Proof By Lemma 17 we just need to show the topology is complete Proposition 37 shows that indprofinite Λmodules are the inverse limit of their discretequotients and hence that the topology on such modules is complete by thecorollary to [1 II Section 35 Proposition 10]
Moreover given indprofinite Λmodules MN the product M timesk N is theinverse limit of discrete modules M prime timesk N prime where M prime and N prime are discretequotients of M and N respectively But M prime timesk N prime = M prime times N prime because bothare discrete so M timesk N = lim
larrminusM prime timesN prime = M timesN the product in the category
of topological modules
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1285
Proposition 39 Suppose that P isin IP (Λ) is projective Then HomTΛ(Pminus)
sends strict exact sequences in PD(Λ) to strict exact sequences in PD(R)
Proof For P profinite this is Corollary 35(vi) For P strictly free P =oplus
Piwe get HomT
Λ(Pminus) =prod
HomTΛ(Piminus) which sends strict exact sequences to
strict exact sequences becauseprod
and HomTΛ(Piminus) do Now the result follows
from Remark 119
Lemma 310 HomTΛ(MN) = HomT
Λop(NlowastMlowast) for all M isin IP (Λ) N isinPD(Λ) naturally in both variables
Proof Think of HomTΛ(MN) and HomT
Λop(NlowastMlowast) as abstract RmodulesThen the functor HomT
Λ(minusQZ) induces maps
HomTΛ(MN)
f1minusrarrHomT
Λ(NlowastMlowast)
f2minusrarrHomT
Λ(NlowastlowastMlowastlowast)
f3minusrarrHomT
Λ(NlowastlowastlowastMlowastlowastlowast)
such that the compositions f2f1 and f3f2 are isomorphisms so f2 is an isomorphism In particular this holds when M is profinite and N is discrete in whichcase the topology on HomT
Λ(MN) is discrete so taking cofinal sequences Mi
for M and N j for N we get HomTΛ(Mi N
j) = HomTΛop(N jlowastMlowast
i ) as topologicalmodules for each i j and the topologies on HomT
Λ(MN) and HomTΛop(NlowastMlowast)
are given by the inverse limits of these Naturality is clear
Corollary 311 Suppose that I isin PD(Λ) is injective Then HomTΛ(minus I)
sends strict exact sequences in IP (Λ) to strict exact sequences in PD(R)
Proposition 312 (Baerrsquos Lemma) Suppose I isin PD(Λ) is discrete Then Iis injective in PD(Λ) if and only if for every closed left ideal J of Λ everymap J rarr I extends to a map Λrarr I
Proof Think of Λ and J as objects of PD(Λ) The condition is clearly necessary To see it is sufficient suppose we are given a strict monomorphismf M rarr N in PD(Λ) and a map g M rarr I Because I is discrete ker(g) isopen in M Because f is strict we can therefore pick an open submodule Uof N such that ker(g) = M cap U So the problem reduces to the discrete caseit is enough to show that M ker(g) rarr I extends to a map NU rarr I In thiscase the proof for abstract modules [14 Baerrsquos Criterion 231] goes throughunchanged
Therefore a discrete Zmodule which is injective in PD(Z) is divisible On the
other hand the discrete Zmodules are just the torsion abelian groups with thediscrete topology So by the version of Baerrsquos Lemma for abstract modules([14 Baerrsquos Criterion 231]) divisible discrete Zmodules are injective in the
category of discrete Zmodules and hence injective in PD(Z) too by Corollary35(iii) So duality gives
Corollary 313 (i) A discrete Zmodule is injective in PD(Z) if and onlyif it is divisible
Documenta Mathematica 21 (2016) 1269ndash1312
1286 Marco Boggi and Ged Corob Cook
(ii) A profinite Zmodule is projective in IP (Z) if and only if it is torsionfree
Proof Being divisible and being torsionfree are Pontryagin dual by [9 Theorem 2912]
Remark 314 On the other hand Qp is not injective in PD(Z) (and hence not
projective in IP (Z) either) despite being divisible (respectively torsionfree)Indeed consider the monomorphism
f Qp rarrprod
N
QpZp x 7rarr (x xp xp2 )
which is strict because its dual
flowast oplus
N
Zp rarr Qp (x0 x1 ) 7rarrsum
n
xnpn
is surjective and hence strict by Proposition 110 Suppose Qp is injective sothat f splits the map g splitting it must send the torsion elements of
prodNQpZp
to 0 because Qp is torsionfree But the torsion elements containoplus
NQpZp
so they are dense inprod
NQpZp and hence g = 0 giving a contradiction
Finally we recall the definition of quasiabelian categories from [10 Definition113] Suppose that E is an additive category with kernels and cokernels Nowf induces a unique canonical map g coim(f)rarr im(f) such that f factors as
Ararr coim(f)gminusrarr im(f)rarr B
and if g is an isomorphism we say f is strict We say E is a quasiabeliancategory if it satisfies the following two conditions
(QA) in any pullback square
Aprimef prime
Bprime
Af
B
if f is strict epic then so is f prime
(QAlowast) in any pushout square
Af
B
Aprimef prime
Bprime
if f is strict monic then so is f prime
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1287
IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pullbacks and Mod(Λ) satisfies (QA) so pullbacks of surjections are surjectionsRecall by Remark 22(ii) that prodiscrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pullbacks Then both categories satisfy axiom(QAlowast) by duality and we have
Proposition 315 IP (Λ) and PD(Λ) are quasiabelian categories
Moreover note that the definition of a strict morphism in a quasiabelian category agrees with our use of the term in IP (Λ) and PD(Λ)
4 Tensor Products
As in the abstract case we can define tensor products of indprofinite modulesSuppose L isin IP (Λop)M isin IP (Λ) N isin IP (R) We call a continuous mapb L timesk M rarr N bilinear if the following conditions hold for all l l1 l2 isinLmm1m2 isinMλ isin Λ
(i) b(l1 + l2m) = b(l1m) + b(l2m)
(ii) b(lm1 +m2) = b(lm1) + b(lm2)
(iii) b(lλm) = b(l λm)
Then T isin IP (R) together with a bilinear map θ L timesk M rarr T is thetensor product of L and M if for every N isin IP (R) and every bilinear mapb L timesk M rarr N there is a unique morphism f T rarr N in IP (R) such thatb = fθIf such a T exists it is clearly unique up to isomorphism and then we writeLotimesΛM for the tensor product To show the existence of LotimesΛM we construct itdirectly b defines a morphism bprime F (LtimeskM)rarr N in IP (R) where F (LtimeskM)is the free indprofinite Rmodule on LtimeskM From the bilinearity of b we getthat the Rsubmodule K of F (Ltimesk M) generated by the elements
(l1 + l2m)minus (l1m)minus (l2m) (lm1 +m2)minus (lm1)minus (lm2) (lλm)minus (l λm)
for all l l1 l2 isin Lmm1m2 isin Mλ isin Λ is mapped to 0 by bprime From thecontinuity of bprime we get that its closure K is mapped to 0 too Thus bprime inducesa morphism bprimeprime F (L timesk M)K rarr N Then it is not hard to check thatF (L timesk M)K together with bprimeprime satisfies the universal property of the tensorproduct
Proposition 41 (i) minusotimesΛminus is an additive bifunctor IP (Λop) times IP (Λ) rarrIP (R)
(ii) There is an isomorphism ΛotimesΛM = M for all M isin IP (Λ) natural in M and similarly LotimesΛΛ = L naturally
Documenta Mathematica 21 (2016) 1269ndash1312
1288 Marco Boggi and Ged Corob Cook
(iii) LotimesΛM = MotimesΛopL naturally in L and M
(iv) Given L in IP (Λop) and M in IP (Λ) with cofinal sequences Li andMj there is an isomorphism
LotimesΛM sim= limminusrarr
IP (R)
(LiotimesΛMj)
Proof (i) and (ii) follow from the universal property
(iii) Writing lowast for the Λopactions a bilinear map bΛ LtimesM rarr N (satisfyingbΛ(lλm) = bΛ(l λm)) is the same thing as a bilinear map bΛop MtimesLrarrN (satisfying bΛop(mλ lowast l) = bΛop(m lowast λ l))
(iv) We have L timesk M = limminusrarr
Li timesMj by Lemma 12 By the universal property of the tensor product the bilinear map lim
minusrarrLi timesMj rarr L timesk M rarr
LotimesΛM factors through f limminusrarr
LiotimesΛMj rarr LotimesΛM and similarly the
bilinear map L timesk M rarr limminusrarr
Li times Mj rarr limminusrarr
LiotimesΛMj factors through
g LotimesΛM rarr limminusrarr
LiotimesΛMj By uniqueness the compositions fg and gfare both identity maps so the two sides are isomorphic
More generally given chain complexes
middot middot middotd1minusrarr L1
d0minusrarr L0dminus1
minusminusrarr middot middot middot
in IP (Λop) and
middot middot middotdprime
1minusrarrM1dprime
0minusrarrM0
dprime
minus1
minusminusrarr middot middot middot
in IP (Λ) both bounded below define the double chain complex LpotimesΛMqwith the obvious vertical maps and with the horizontal maps defined in theobvious way except that they are multiplied by minus1 whenever q is odd thismakes Tot(LotimesΛM) into a chain complex which we denote by LotimesΛM Eachterm in the total complex is the sum of finitely many indprofinite Rmodulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)
Suppose from now on that ΘΦΨ are profinite Ralgebras Then let IP (ΘminusΦ) be the category of indprofinite Θ minus Φbimodules and Θ minus Φkbimodulehomomorphisms We leave the details to the reader after noting that an indprofinite Rmodule N with a left Θaction and a right Φaction which arecontinuous on profinite submodules is an indprofinite Θ minus Φbimodule sincewe can replace a cofinal sequence Ni of profinite Rmodules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦbimodules If L is an indprofinite ΘminusΛbimodule and M is an indprofinite ΛminusΦbimodule one can make LotimesΛM intoan indprofinite Θminus Φbimodule in the same way as in the abstract case
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1289
Theorem 42 (Adjunction isomorphism) Suppose L isin IP (Θ minus Λ)M isinIP (Λminus Φ) N isin PD(ΘminusΨ) Then there is an isomorphism
HomTΘ(LotimesΛMN) sim= HomT
Λ(MHomTΘ(LN))
in PD(ΦminusΨ) natural in LMN
Proof Given cofinal sequences Li Mj Nk in LMN respectively we
have natural isomorphisms
HomTΘ(LiotimesΛMj Nk) sim= HomT
Λ(Mj HomTΘ(Li Nk))
of discrete Φ minus Ψbimodules for each i j k by [9 Proposition 554(c)] Thenby Lemma 212 we have
HomTΘ(LotimesΛMN) sim= lim
larrminusPD(ΦminusΨ)
HomTΘ(LiotimesΛMj Nk)
sim= limlarrminus
PD(ΦminusΨ)
HomTΛ(Mj Hom
TΘ(Li Nk))
sim= HomTΛ(MHomT
Θ(LN))
It follows that HomTΛ (considered as a cocovariant bifunctor IP (Λ)op times
PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ commutes with colimits in both variables by [14 Theorem 2610]
If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a prodiscreteΦminusΘbimodule and similarly with indprofinite and prodiscrete switched
Corollary 43 There is a natural isomorphism
(LotimesΛM)lowast sim= HomTΛ(MLlowast)
in PD(ΦminusΘ) for L isin IP (Θminus Λ)M isin IP (Λminus Φ)
Proof Apply the theorem with Ψ = Z and N = QZ
Properties proved about HomΛ in the past two sections carry over immediatelyto properties of otimesΛ using this natural isomorphism Details are left to thereaderGiven a chain complex M in IP (Λ) and a cochain complex N in PD(Λ)both bounded below if we apply lowast to the double complex with (p q)th termHomT
Λ(Mp Nq) we get a double complex with (q p)th term N qlowastotimesΛMp ndash
note that the indices are switched This changes the sign convention usedin forming HomT
Λ(MN) into the one used in forming NlowastotimesΛM and so we haveHomT
Λ(MN)lowast = NlowastotimesΛM (because lowast commutes with finite direct sums)
Documenta Mathematica 21 (2016) 1269ndash1312
1290 Marco Boggi and Ged Corob Cook
5 Derived Functors in QuasiAbelian Categories
We give a brief sketch of the machinery needed to derive functors in quasiabelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasiabeliancategory unless otherwise stated dn will be the map An+1 rarr An Dually if(A d) is a cochain complex dn will be the map An rarr An+1Given a quasiabelian category E let K(E) be the category whose objects arecochain complexes in E and whose morphisms are maps of cochain complexesup to homotopy this makes K(E) into a triangulated category Given a cochaincomplex A in E we say A is strict exact in degree n if the map dnminus1 Anminus1 rarrAn is strict and im(dnminus1) = ker(dn) We say A is strict exact if it is strictexact in degree n for all n Then writing N(E) for the full subcategory ofK(E) whose objects are strict exact we get that N(E) is a null system so wecan localise K(E) at N(E) to get the derived category D(E) We also defineK+(E) to be the full subcategory of K(E) whose objects are bounded belowand Kminus(E) to be the full subcategory whose objects are bounded above wewrite D+(E) and Dminus(E) for their localisations respectively We say a map ofcomplexes in K(E) is a strict quasiisomorphism if its cone is in N(E)Deriving functors in quasiabelian categories uses the machinery of tstructuresThis can be thought of as giving a wellbehaved cohomology functor to a triangulated category For more detail on tstructures see [8 Section 13]Given a triangulated category T with translation functor T a tstructure on Tis a pair T le0 T ge0 of full subcategories of T satisfying the following conditions
(i) T (T le0) sube T le0 and Tminus1(T ge0) sube T ge0
(ii) HomT (XY ) = 0 for X isin T le0 Y isin Tminus1(T ge0)
(iii) for all X isin T there is a distinguished triangle X0 rarr X rarr X1 rarr withX0 isin T
le0 X1 isin Tminus1(T ge0)
It follows from this definition that if T le0 T ge0 is a tstructure on T there isa canonical functor τle0 T rarr T le0 which is left adjoint to inclusion and acanonical functor τge0 T rarr T ge0 which is right adjoint to inclusion One canthen define the heart of the tstructure to be the full subcategory T le0 cap T ge0and the 0th cohomology functor
H0 T rarr T le0 cap T ge0
by H0 = τge0τle0
Theorem 51 The heart of a tstructure on a triangulated category is anabelian category
There are two canonical tstructures on D(E) the left tstructure and the righttstructure and correspondingly a left heart LH(E) and a right heart RH(E)The tstructures and hearts are dual to each other in the sense that there is
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1291
a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop
is quasiabelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left tstructure on D(E) is given by taking T le0 to be the complexes which are strict exact in all positive degrees and T ge0 to be the complexes which are strict exact in all negative degrees LH(E) is therefore thefull subcategory of D(E) whose objects are strict exact in every degree except0 the 0th cohomology functor
LH0 D(E)rarr LH(E)
is given by0rarr coim(dminus1)rarr ker(d0)rarr 0
Every object of LH(E) is isomorphic to a complex
0rarr Eminus1 fminusrarr E0 rarr 0
of E with E0 in degree 0 and f monic Let I E rarr LH(E) be the functor givenby
E 7rarr (0rarr E rarr 0)
with E in degree 0 Let C LH(E)rarr E be the functor given by
(0rarr Eminus1 fminusrarr E0 rarr 0) 7rarr coker(f)
Proposition 52 I is fully faithful and right adjoint to C In particular identifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence
0rarr LrarrM rarr N rarr 0
in E its image under I is a short exact sequence in LH(E) if and only if thesequence is short strict exact in E
The functor I induces a functor D(I) D(E)rarr D(LH(E))
Proposition 53 D(I) is an equivalence of categories which exchanges the lefttstructure of D(E) with the standard tstructure of D(LH(E)) This inducesequivalences D(E)+ rarr D(LH(E))+ and D(E)minus rarr D(LH(E))minus
Thus there are cohomological functors LHn D(E)rarr LH(E) so that given anydistinguished triangle in D(E) we get long exact sequences in LH(E) Givenan object (A d) isin D(E) LHn(A) is the complex
0rarr coim(dnminus1)rarr ker(dn)rarr 0
with ker(dn) in degree 0Everything for RH(E) is done dually so in particular we get
Documenta Mathematica 21 (2016) 1269ndash1312
1292 Marco Boggi and Ged Corob Cook
Lemma 54 The functors RHn D(Eop)rarrRH(Eop) are given by
(LHminusn)op D(E)op rarrRH(E)op
As for PD(Λ) we say an object I of E is injective if for any strict monomorphism E rarr Eprime in E any morphism E rarr I extends to a morphism Eprime rarr I andwe say E has enough injectives if for everyE isin E there is a strict monomorphismE rarr I for some injective I
Proposition 55 The right heart RH(E) of E has enough injectives if andonly if E does An object I isin E is injective in E if and only if it is injective inRH(E)
Suppose that E has enough injectives Write I for the full subcategory of Ewhose objects are injective in E
Proposition 56 Localisation at N+(I) gives an equivalence of categoriesK+(I)rarr D+(E)
We can now define derived functors in the same way as the abelian case Suppose we are given an additive functor F E rarr E prime between quasiabelian categories Let Q K+(E) rarr D+(E) and Qprime K+(E prime) rarr D+(E prime) be the canonicalfunctors Then the right derived functor of F is a triangulated functor
RF D+(E)rarr D+(E prime)
(that is a functor compatible with the triangulated structure) together with anatural transformation
t Qprime K+(F )rarr RF Q
satisfying the property that given another triangulated functor
G D+(E)rarr D+(E prime)
and a natural transformation
g Qprime K+(F )rarr G Q
there is a unique natural transformation h RF rarr G such that g = (h Q)tClearly if RF exists it is unique up to natural isomorphismSuppose we are given an additive functor F E rarr E prime between quasiabeliancategories and suppose E has enough injectives
Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasiisomorphism E rarr I such that each In is injective and each En rarr In is a strictmonomorphism
We say such an I is an injective resolution of E
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1293
Proposition 58 In the situation above the right derived functor of F existsand RF (E) = K+(F )(I) for any injective resolution I of E
We write RnF for the composition RHn RF
Remark 59 Since RF is a triangulated functor we could also define the cohomological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct CartanEilenberg resolutions in this category and hence prove a Grothendieck spectralsequence Theorem 512 below On the other hand it is not clear that such aspectral sequence holds for LHnRF and in this sense RHnRF is the lsquorightrsquodefinition ndash but see Lemma 68
The construction of derived functors generalises to the case of additive bifunctors F E times E prime rarr E primeprime where E and E prime have enough injectives the right derivedfunctor
RF D+(E)timesD+(E prime)rarr D+(E primeprime)
exists and is given by RF (EEprime) = sK+(F )(I I prime) where I I prime are injective resolutions of EEprime and sK+(F )(I I prime) is the total complex of the double complexK+(F )(Ip I primeq)pq in which the vertical maps with p odd are multiplied byminus1Projectives are defined dually to injectives left derived functors are defineddually to right derived ones and if a quasiabelian category E has enoughprojectives then an additive functor F from E to another quasiabelian categoryhas a left derived functor LF which can be calculated by taking projectiveresolutions and we write LnF for LHminusn LF Similarly for bifunctorsWe state here for future reference some results on spectral sequences see [14Chapter 5] for more details All of the following results have dual versionsobtained by passing to the opposite category and we will use these dual resultsinterchangeably with the originals Suppose that A = Apq is a bounded belowdouble cochain complex in E that is there are only finitely many nonzeroterms on each diagonal n = p + q and the total complex Tot(A) is boundedbelow By Proposition 53 we can equivalently think of A as a bounded belowdouble complex in the abelian category RH(E) Then we can use the usualspectral sequences for double complexes
Proposition 510 There are two bounded spectral sequences
IEpq2 = RHp
hRHqv (A)
IIEpq2 = RHp
vRHqh(A)
rArr RHp+q Tot(A)
naturally in A
Proof [14 Section 56]
Suppose we are given an additive functor F E rarr E prime between quasiabeliancategories and consider the case where A isin D+(E) Suppose E has enoughinjectives so that RH(E) does too Thinking of A as an object in D+(RH(E))
Documenta Mathematica 21 (2016) 1269ndash1312
1294 Marco Boggi and Ged Corob Cook
we can take a bounded below CartanEilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result
Proposition 511 There are two bounded spectral sequences
IEpq2 = RHp(RqF (A))
IIEpq2 = (RpF )(RHq(A))
rArr Rp+qF (A)
naturally in A
Proof [14 Section 57]
Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasiabelian categories where E and E prime have enough injectivesSuppose G sends injective objects of E to injective objects of E prime
Theorem 512 (Grothendieck Spectral Sequence) For A isin D+(E) there isa natural isomorphism R(FG)(A) rarr (RF )(RG)(A) and a bounded spectralsequence
IEpq2 = (RpF )(RqG(A))rArr Rp+q(FG)(A)
naturally in A
Proof Let I be an injective resolution of A There is a natural transformationR(FG) rarr (RF )(RG) by the universal property of derived functors it is anisomorphism because by hypothesis each G(In) is injective and hence
(RF )(RG)(A) = F (G(I)) = R(FG)(A)
For the spectral sequence apply Proposition 511 with A = G(I) We have
IEpq2 = RHp(RqF (G(I)))rArr Rp+qF (G(I))
by the injectivity of the G(In) RqF (G(I)) = 0 for q gt 0 so the spectralsequence collapses to give
Rp+qF (G(I)) sim= RHp(FG(I)) = Rp(FG)(A)
On the other hand
IIEpq2 = (RpF )(RHq(G(I))) = (RpF )(RqG(A))
and the result follows
We consider once more the case of an additive bifunctor
F E times E prime rarr E primeprime
for E E prime and E primeprime quasiabelian this induces a triangulated functor
K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1295
in the sense that a distinguished triangle in one of the variables and a fixedobject in the other maps to a distinguished triangle in K+(E primeprime) Hence for afixed A isin K+(E) K+(F ) restricts to a triangulated functor K+(F )(Aminus) andif E prime has enough injectives we can derive this to get a triangulated functor
R(F (Aminus)) D+(E prime)rarr D+(E primeprime)
Maps A rarr Aprime in K+(E) induce natural transformations R(F (Aminus)) rarrR(F (Aprimeminus)) so in fact we get a functor which we denote by
R2F K+(E)timesD+(E prime)rarr D+(E primeprime)
We know R2F is triangulated in the second variable and it is triangulated inthe first variable too because given B isin D+(E prime) with an injective resolution IR2F (minus B) = K+(F )(minus I) is a triangulated functor K+(E)rarr D+(E primeprime)Similarly we can define a triangulated functor
R1F D+(E)timesK+(E prime)rarr D+(E primeprime)
by deriving in the first variable if E has enough injectives
Proposition 513 (i) If E prime has enough injectives and F (minus J) E rarr E primeprime isstrict exact for J injective then R2F (minus B) sends quasiisomorphismsto isomorphisms that is we can think of R2F as a functor D+(E) timesD+(E prime)rarr D+(E primeprime)
(ii) Suppose in addition that E has enough injectives Then R2F is naturallyisomorphic to RF
Similarly with the variables switched
Proof (i) R2F (minus B) = K+(F )(minus I) for an injective resolution I of BGiven a quasiisomorphism A rarr Aprime in K+(E) consider the map of double complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasiisomorphism of the correspondingtotal complexes
(ii) This holds by the same argument as (i) taking Aprime to be an injectiveresolution of A
6 Derived Functors in IP (Λ) and PD(Λ)
We now use the framework of Section 5 to define derived functors in our categories of interest Note first that the dual equivalence between IP (Λ) andPD(Λ) extends to dual equivalences between Dminus(IP (Λ)) and D+(PD(Λ))given by applying the functor lowast to cochain complexes in these categoriesby defining (Alowast)n = (Aminusn)lowast for a cochain complex A in PD(Λ) and similarly for the maps We will also identify Dminus(IP (Λ)) with the category of
Documenta Mathematica 21 (2016) 1269ndash1312
1296 Marco Boggi and Ged Corob Cook
chain complexes A (localised over the strict quasiisomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality extends to one between LH(IP (Λ)) and RH(PD(Λ)) Moreover writing RHn
and LHn for the nth cohomological functors D(PD(R)) rarr RH(PD(R)) andD(IP (Rop))rarr LH(IP (Rop)) respectively the following is just a restatementof Lemma 54
Lemma 61 LHminusn lowast = lowast RHn
Let
RHomTΛ(minusminus) D
minus(IP (Λ))timesD+(PD(Λ))rarr D+(PD(R))
be the right derived functor of HomTΛ(minusminus) IP (Λ) times PD(Λ) rarr PD(R) By
Proposition 58 this exists because IP (Λ) has enough projectives and PD(Λ)has enough injectives and RHomT
Λ(MN) is given by HomTΛ(P I) where P is
a projective resolution of M and I is an injective resolution of N Dually let
minusotimesLΛminus Dminus(IP (Λop))timesDminus(IP (Λ))rarr Dminus(IP (R))
be the left derived functor of minusotimesΛminus IP (Λop) times IP (Λ) rarr IP (R) Then by
Proposition 58 again MotimesLΛN is given by P otimesΛQ where PQ are projective
resolutions of MN respectivelyWe also define ExtnΛ to be the composite
LH(IP (Λ))timesRH(PD(Λ))rarr Dminus(IP (Λ))timesD+(PD(Λ))
RHomTΛminusminusminusminusminusrarr D+(PD(R))
RHn
minusminusminusrarr RH(PD(R))
and TorΛn to be the composite
LH(IP (Λop))times LH(IP (Λ))rarr Dminus(IP (Λop))times Dminus(IP (Λ))
otimesL
Λminusminusrarr Dminus(IP (R))
LHminusn
minusminusminusminusrarr LH(IP (R))
where in both cases the unlabelled maps are the obvious inclusions of fullsubcategories Because LHn and RHn are cohomological functors we get theusual long exact sequences in LH(IP (R)) and RH(PD(R)) coming from strictshort exact sequences (in the appropriate category) in either variable naturalin both variables ndash since these give distinguished triangles in the correspondingderived categoryWhen ExtnΛ or TorΛn take coefficients in IP (Λ) or PD(Λ) these coefficientmodules should be thought of as objects in the appropriate left or right heartvia inclusion of the full subcategory
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1297
Remark 62 The reason we cannot define lsquoclassicalrsquo derived functors in thesense of say [14] just in terms of topological module categories is essentiallythat these categories like most interesting categories of topological modulesfail to be abelian Intuitively this means that the naive definition of the homology of a chain complex of such modules ndash that is defining
Hn(M) = coker(coim(dnminus1)rarr ker(dn))
ndash loses too much information There is no wellbehaved homology functor fromchain complexes in a quasiabelian category back to the category itself so thata naive approach here fails That is why we must use the more sophisticatedmachinery of passing to the left or right hearts which function as lsquocompletionsrsquoof the original category to an abelian category in an appropriate sense see [10]Our Ext and Tor functors are the appropriate analogues in this setting of theclassical derived functors in a sense made precise in the following proposition
Recall from Proposition 53 that we have equivalences Dminus(IP (Λ)) rarrDminus(LH(IP (Λ))) and D+(PD(Λ)) rarr D+(LH(PD(Λ))) So we may think ofRHomT
Λ(minusminus) as a functor
Dminus(LH(IP (Λ))) timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
via these equivalences and similarly for otimesLΛ For the next proposition we use
these definitionsNote that by Remark 611 below Ext0Λ(minusminus) 6= HomT
Λ(minusminus) as functors onIP (Λ)times PD(Λ)
Proposition 63 RHomTΛ(minusminus) and ExtnΛ(minusminus) are respectively the total
derived functor and the nth classical derived functor of Ext0Λ(minusminus) Similarly
otimesLΛ and TorΛn(minusminus) are respectively the total derived functor and the nth
classical derived functor of TorΛ0 (minusminus)
Proof We prove the first statement the second can be shown similarly Write
RExt0Λ(minusminus) Dminus(LH(IP (Λ)))timesD+(LH(PD(Λ)))rarr D+(LH(PD(R)))
for the total right derived functor of Ext0Λ(minusminus) Then for M isinDminus(LH(IP (Λ))) with projective resolution P and N isin D+(LH(PD(Λ))) withinjective resolution I RExt0Λ(MN) is by definition the total complex of thebicomplex (Ext0Λ(Pp I
q))pq But Ext0Λ(Pp Iq) = HomT
Λ(Pp Iq) because Pp is
projective and so is a resolution of itself So the bicomplex is (HomTΛ(Pp I
q))pq
and its total complex by definition is RHomTΛ(MN) giving the result for
total derived functors Taking M isin LH(IP (Λ)) and N isin LH(PD(Λ))we get that the nth classical derived functor is RHn RExt0Λ(MN) =RHn RHomT
Λ(MN) = ExtnΛ(MN)
Lemma 64 (i) RHomTΛ(minusminus) and otimes
LΛ are Pontryagin dual in the sense
that given M isin Dminus(IP (Λ)) and N isin D+(PD(Λ)) there holds
RHomTΛ(MN)lowast = Nlowastotimes
LΛM naturally in MN
Documenta Mathematica 21 (2016) 1269ndash1312
1298 Marco Boggi and Ged Corob Cook
(ii) For M isin LH(IP (Λ)) and N isin RH(PD(Λ)) ExtnΛ(MN)lowast =TorΛn(N
lowastM)
Proof We prove (i) (ii) follows by taking cohomology Take a projective resolution P of M and an injective resolution I of N so that by duality Ilowast is aprojective resolution of Nlowast Then
RHomTΛ(MN)lowast = HomT
Λ(P I)lowast = IlowastotimesΛP = Nlowastotimes
LΛM
naturally by the universal property of derived functors
Remark 65 More generally as functors on the appropriate categories of bi
modules it follows from Theorem 42 that LotimesLΛminus is left adjoint to the functor
RHomTΘ(Lminus) for L isin IP (ΘminusΛ) and similarly for the Ext and Tor functors
Details are left to the reader
Proposition 66 (i) RHomTΛ(MN) = RHomT
Λop(Mlowast Nlowast) andExtnΛ(MN) = ExtnΛop(NlowastMlowast)
(ii) NlowastotimesLΛM = Motimes
LΛopNlowast and TorΛn(N
lowastM) = TorΛop
n (MNlowast)naturally in MN
Proof (ii) follows from (i) by Pontryagin duality To see (i) take a projectiveresolution P of M and an injective resolution I of N Then
RHomTΛ(MN) = HomT
Λ(P I) = HomTΛop(Ilowast P lowast) = RHomT
Λop(Mlowast Nlowast)
by Lemma 310 The rest follows by applying LHminusn
Proposition 67 RHomTΛ Ext otimes
LΛ and Tor can be calculated using a reso
lution of either variable That is given M with a projective resolution P andN with an injective resolution I in the appropriate categories
RHomTΛ(MN) = HomT
Λ(PN) = HomTΛ(M I)
ExtnΛ(MN) = RHn(HomTΛ(PN)) = Hn(HomT
Λ(M I))
NlowastotimesLΛM = NlowastotimesΛP = IlowastotimesΛM and
TorΛn(NlowastM) = LHminusn(NlowastotimesΛP ) = LHminusn(IlowastotimesΛM)
Proof By Proposition 513 RHomTΛ(MN) = HomT
Λ(M I) everything elsefollows by some combination of Proposition 66 taking cohomology and applying Pontryagin duality
We will now see that for moduletheoretic purposes it is sometimes moreuseful to apply LHn to right derived functors and RHn to left derived functorsthough as noted in Remark 59 the resulting cohomological functors are notso wellbehaved
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1299
Lemma 68 LH0 RHomTΛ(MN) = HomT
Λ(MN) and RH0(NlowastotimesLΛM) =
NlowastotimesΛM for all M isin IP (Λ) N isin PD(Λ) naturally in MN
Proof Take a projective resolution P of M Then
LH0 RHomTΛ(MN) = ker(HomT
Λ(P0 N)rarr HomTΛ(P1 N))
= HomTΛ(ker(P0 rarr P1) N)
= HomTΛ(MN)
because HomTΛ commutes with kernels The rest follows by duality
Example 69 Zp is projective in IP (Z) by Corollary 313 Now consider thesequence
0rarroplus
N
Zpfminusrarr
oplus
N
Zpgminusrarr Qp rarr 0
where f is given by (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 ) and g isgiven by (x0 x1 x2 ) 7rarr x0 + x1p+ x2p
2 + middot middot middot This sequence is exact onthe underlying modules so by Proposition 110 it is strict exact and hence itis a projective resolution of Qp By applying Pontryagin duality we also getan injective resolution
0rarr Qp rarrprod
N
QpZp rarrprod
N
QpZp rarr 0
Recall that by Remark 314 Qp is not projective or injective
Lemma 610 For all n gt 0 and all M isin IP (Z)
(i) ExtnZ(QpM
lowast) = 0
(ii) ExtnZ(MQp) = 0
(iii) TorZn(QpM) = 0
(iv) TorZn(MlowastQp) = 0
Proof By Lemma 64 and Proposition 66 it is enough to prove (iii) Since Qp
has a projective resolution of length 1 the statement is clear for n gt 1 Now
TorZ1 (QpM) = ker(fotimesZM) in the notation of the example Writing Mp for
ZpotimesZM fotimes
ZM is given by
oplus
N
Mp rarroplus
N
Mp (x0 x1 x2 ) 7rarr (x0 x1 minus p middot x0 x2 minus p middot x1 )
because otimesZcommutes with direct sums But this map is clearly injective as
required
Documenta Mathematica 21 (2016) 1269ndash1312
1300 Marco Boggi and Ged Corob Cook
Remark 611 By Lemma 610 Ext0Z(Qpminus) is an exact functor from the
category RH(PD(Z)) to itself In particular writing I for the inclusion
functor PD(Z) rarr RH(PD(Z)) the composite Ext0Z(Qpminus) I sends short
strict exact sequences in PD(Z) to short exact sequences in RH(PD(Z)) byProposition 52 On the other hand by Proposition 52 again the compositeI HomT
Z(Qpminus) does not send short strict exact sequences in PD(Z) to short
exact sequences in RH(PD(Z)) Therefore by [10 Proposition 1310] and inthe terminology of [10] HomT
Z(Qpminus) is not RR left exact there is some short
strict exact sequence0rarr LrarrM rarr N rarr 0
in PD(Z) such that the induced map HomTZ(QpM) rarr HomT
Z(Qp N) is not
strict By duality a similar result holds for tensor products with Qp
7 Homology and cohomology of profinite groups
Let G be a profinite group We define the category of indprofinite right Gmodules IP (Gop) to have as its objects indprofinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homomorphisms which are compatible with the Gaction We define the category ofprodiscrete Gmodules PD(G) to have as its objects prodiscrete ZmodulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the GactionFrom now on R will denote a commutative profinite ring
Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent
(ii) An indprofinite right RJGKmodule is the same as an indprofinite Rmodule M with a continuous map MtimeskGrarrM such that (mr)g = (mg)rfor all g isin G r isin Rm isinM
(iii) PD(G) and PD(ZJGK) are equivalent
(iv) A prodiscrete RJGKmodule is the same as a prodiscrete Rmodule Mwith a continuous map G timesM rarr M such that g(rm) = r(gm) for allg isin G r isin Rm isinM
Proof (i) Given M isin IP (Gop) take a cofinal sequence Mi for M as anindprofinite abelian group Replacing each Mi with M prime
i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightGmodules By [9 Proposition 536(c)] each M prime
i canonically has the
structure of a profinite right ZJGKmodule and with this structure the
cofinal sequence M primei makes M into an object in IP (ZJGKop) This
gives a functor IP (Gop) rarr IP (ZJGKop) Similarly we get a functor
IP (ZJGKop) rarr IP (Gop) by taking cofinal sequences and forgetting the
Zstructure on the profinite elements in the sequence These functors areclearly inverse to each other
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1301
(ii) Similarly
(iii) Similarly replacing [9 Proposition 536(c)] with [9 Proposition 536(e)]
(iv) Similarly
By (ii) of Proposition 71 given M isin IP (R) we can think of M as an objectin IP (RJGKop) with trivial Gaction This gives a functor the trivial modulefunctor IP (R)rarr IP (RJGKop) which clearly preserves strict exact sequencesGiven M isin IP (RJGKop) define the coinvariant module MG by
M〈m middot g minusm for all g isin Gm isinM〉
This makes MG into an object in IP (R) In the same way as for abstractmodules MG is the maximal quotient module of M with trivial Gaction andso minusG becomes a functor IP (RJGKop) rarr IP (R) which is left adjoint to thetrivial module functor We can define minusG similarly for left indprofinite RJGKmodulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial Gaction This gives a functor which we also call thetrivial module functor PD(R) rarr PD(RJGK) which clearly preserves strictexact sequencesGiven M isin PD(RJGK) define the invariant submodule MG by
m isinM g middotm = m for all g isin Gm isinM
It is a closed submodule of M because
MG =⋂
gisinG
ker(M rarrMm 7rarr g middotmminusm)
Therefore we can think of MG as an object in PD(R) In the same way as forabstract modules MG is the maximal submodule of M with trivial Gactionand so minusG becomes a functor PD(RJGK) rarr PD(R) which is right adjoint tothe trivial module functor We can define minusG similarly for right prodiscreteRJGKmodules
Lemma 72 (i) For M isin IP (RJGKop) MG = MotimesRJGKR
(ii) For M isin PD(RJGK) MG = HomTRJGK(RM)
Proof (i) Pick a cofinal sequence Mi forM By [9 Lemma 633] (Mi)G =MiotimesRJGKR naturally in Mi As a left adjoint minusG commutes with directlimits so
MG = limminusrarr
(Mi)G = limminusrarr
(MiotimesRJGKR) = MotimesRJGKR
by Proposition 41
Documenta Mathematica 21 (2016) 1269ndash1312
1302 Marco Boggi and Ged Corob Cook
(ii) Similarly by [9 Lemma 621] because minusG and HomTRJGK(Rminus) commute
with inverse limits
Corollary 73 Given M isin IP (RJGKop) (MG)lowast = (Mlowast)G
Proof Lemma 72 and Corollary 43
We now define the nth homology functor of G over R by
HRn (Gminus) = TorRJGK
n (minus R) LH(IP (RJGKop))rarr LH(IP (R))
and the nth cohomology functor of G over R by
HnR(Gminus) = ExtnRJGK(Rminus) RH(PD(RJGK))rarrRH(PD(R))
As noted in Remark 62 we can also think of HRn (Gminus) as a functor
IP (RJGKop)rarr LH(IP (R))
by precomposing with inclusion from these subcategories and we may do sowithout further comment
We have by Lemma 64 that
Proposition 74 HRn (GM)lowast = Hn
R(GMlowast) for all M isin LH(IP (RJGKop))naturally in M
Of course one can calculate all these objects using the projective resolutionof R arising from the usual bar resolution [9 Section 62] and this showsthat the homology and cohomology are unchanged if we forget the Rmodulestructure and think of M as an object of LH(IP (ZJGKop)) that is the underlying complex of abelian kgroups of HR
n (GM) and the underlying complex
of topological abelian groups of HnR(GMlowast) are H Z
n(GM) and HnZ(GMlowast)
respectively We therefore write
Hn(GM) = H Z
n(GM) and
Hn(GMlowast) = HnZ(GMlowast)
Theorem 75 (Universal Coefficient Theorem) Suppose M isin PD(ZJGK) hastrivial Gaction Then there are noncanonically split short strict exact sequences
0rarr Ext1Z(Hnminus1(G Z)M)rarr Hn(GM)rarr Ext0
Z(Hn(G Z)M)rarr 0
0rarr TorZ0(Mlowast Hn(G Z))rarr Hn(GMlowast)rarr TorZ1 (M
lowast Hnminus1(G Z))rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1303
Proof We prove the first sequence the second follows by Pontryagin dualityTake a projective resolution P of Z in IP (ZJGK) with each Pn profinite sothat Hn(GM) = Hn(HomT
ZJGK(PM)) Because M has trivial Gaction M =
HomTZ(ZM) where we think of Z as an indprofinite Zminus ZJGKbimodule with
trivial Gaction So
HomTZJGK
(PM) = HomTZJGK
(PHomTZ(ZM))
= HomTZ(Zotimes
ZJGKPM)
= HomTZ(PGM)
Note that PG is a complex of profinite modules so all the maps involved areautomatically strict Since minusG is left adjoint to an exact functor (the trivialmodule functor) we get in the same way as for abelian categories that minusG
preserves projectives so each (Pn)G is projective in IP (Z) and hence torsionfree by Corollary 313 Now the profinite subgroups of each (Pn)G consisting
of cycles and boundaries are torsionfree and hence projective in IP (Z) byCorollary 313 so PG splits Then the result follows by the same proof as inthe abstract case [14 Section 36]
Corollary 76 For all n
(i) Hn(GZp) = TorZ0 (Zp Hn(G Z)) = ZpotimesZHn(G Z)
(ii) Hn(GQp) = TorZ0 (Qp Hn(G Z))
(iii) Hn(GQp) = Ext0Z(Hn(G Z)Qp)
Proof (i) holds because Zp is projective (ii) and (iii) follow from Lemma610
Suppose now that H is a (profinite) subgroup of G We can think of RJGKas an indprofinite RJHK minus RJGKbimodule the left Haction is given by leftmultiplication byH onG and the rightGaction is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the Gaction to an Haction Moreovermaps of Gmodules which are compatible with the Gaction are compatiblewith the Haction So restriction gives a functor
ResGH IP (RJGK)rarr IP (RJHK)
ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Similarly we can define a restriction functor
ResGH PD(RJGKop)rarr PD(RJHKop)
by HomTRJGK(RJHցGւGKminus)
Documenta Mathematica 21 (2016) 1269ndash1312
1304 Marco Boggi and Ged Corob Cook
On the other hand given M isin IP (RJHKop) MotimesRJHKRJHցGւGK becomes an
object in IP (RJGKop) In this way minusotimesRJHKRJHցGւGK becomes a functorinduction
IndGH IP (RJHKop)rarr IP (RJGKop)
Also HomTRJHK(RJHցGւGKminus) becomes a functor coinduction which we de
note by
CoindGH PD(RJHK)rarr PD(RJGK)
Since RJHցGւGK is projective in IP (RJHK) and IP (RJGK)op ResGH IndGH andCoindGH all preserve strict exact sequences Moreover ResGH and IndGH commutewith colimits of indprofinite modules because tensor products do and ResGHand CoindGH commute with limits of prodiscrete modules because Hom doesin the second variableWe can similarly define restriction on right indprofinite or left prodiscreteRJGKmodules induction on left indprofinite RJGKmodules and coinductionon right prodiscrete RJGKmodules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left Haction together with a topology thatmakes it into both an indprofinite Hmodule and a prodiscrete HmoduleFor example this is the case if M is secondcountable profinite or countablediscrete Then both IndGH and CoindGH are defined When H is open in Gwe get IndG
H minus = CoindGH minus in the same way as the abstract case [14 Lemma634]
Lemma 77 For M isin IP (RJHKop) (IndGH M)lowast = CoindGH(Mlowast) For N isin
IP (RJGKop) (ResGH N)lowast = ResGH(Nlowast)
Proof
(IndGH M)lowast = (MotimesRJHKRJHցGւGK)lowast
= HomTRJHK(RJHցGւGKMlowast) = CoindGH(Mlowast)
(ResGH N)lowast = (NotimesRJGKRJGցGւHK)lowast
= HomTRJHK(RJGցGւHK Nlowast) = ResGH(Nlowast)
Lemma 78 (i) IndGH is left adjoint to ResGH That is for M isin IP (RJHK)N isin IP (RJGK) HomIP
RJGK(IndGH MN) = HomIP
RJHK(MResGH N) naturally in M and N
(ii) CoindGH is right adjoint to ResGH That is for M isin PD(RJGK) N isinPD(RJHK) HomPD
RJGK(MCoindGH N) = HomPDRJHK(Res
GH MN) natu
rally in M and N
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1305
Proof (i) and (ii) are equivalent by Pontryagin duality and Lemma 77 Weshow (i) Pick cofinal sequences Mi Nj for MN Then
HomIPRJGK(Ind
GH MN) = HomIP
RJGK(limminusrarr(IndG
H Mi) limminusrarrNj)
= limlarrminusi
limminusrarrj
HomIPRJGK(Ind
GH Mi Nj)
= limlarrminusi
limminusrarrj
HomIPRJHK(MiRes
GH Nj)
= HomIPRJHK(limminusrarr
Mi limminusrarrResGH Nj)
= HomIPRJHK(MResGH N)
by Lemma 115 and the Pontryagin dual of [9 Lemma 6102] and all theisomorphisms in this sequence are natural
Corollary 79 The functor IndGH sends projectives in IP (RJHK) to projec
tives in IP (RJGK) Dually CoindGH sends injectives in PD(RJHK) to injectivesin PD(RJGK)
Proof The adjunction of Lemma 78 shows that for P isin IP (RJHK) projective HomIP
RJGK(IndGH Pminus) = HomIP
RJHK(PResGH minus) sends strict epimorphisms
to surjections as required
The second statement follows from the first by applying the result for IndGH to
IP (RJHKop) and then using Pontryagin duality
Lemma 710 For M isin IP (RJHKop) N isin IP (RJGK) IndGH MotimesRJGKN =
MotimesRJHK ResGH N and HomT
RJGK(NCoindGH(Mlowast)) = HomTRJHK(NMlowast) natu
rally in MN
Proof IndGH MotimesRJGKN = MotimesRJHKRJHցGւGKotimesRJGKN = MotimesRJHK Res
GH N
The second equation follows by applying Pontryagin duality and Lemma 77
Theorem 711 (Shapirorsquos Lemma) For M isin Dminus(IP (RJHKop)) N isinDminus(IP (RJGK)) we have
(i) IndGH MotimesLRJGKN = Motimes
LRJHK Res
GH N
(ii) NotimesLRJGKop Ind
GH M = ResGH Notimes
LRJHKopM
(iii) RHomTRJGKop(Ind
GH MNlowast) = RHomT
RJHKop(MResGH Nlowast)
(iv) RHomTRJGK(NCoindGH Mlowast) = RHomT
RJHK(ResGH NMlowast)
naturally in MN Similar statements hold for the Ext and Tor functors
Documenta Mathematica 21 (2016) 1269ndash1312
1306 Marco Boggi and Ged Corob Cook
Proof We show (i) (ii)(iv) follow by Lemma 64 and Proposition 66 Take aprojective resolution P of M By Corollary 79 IndG
H P is a projective resolution of IndG
H M Then
IndGH MotimesLRJGKN = IndGH P otimesRJGKN
= P otimesRJHK ResGH N by Lemma 710
= MotimesLRJHK Res
GH N
and all these isomorphisms are natural For the rest apply the cohomologyfunctors
Corollary 712 For M isin IP (RJHKop)
HRn (G IndGH M) = HR
n (HM) and
HnR(GCoindGH Mlowast) = Hn
R(HMlowast)
for all n naturally in M
Proof Apply Shapirorsquos Lemma with N = R with trivial Gaction ndash the restriction of this action to H is also trivial
If K is a profinite normal subgroup of G then for M isin IP (RJGKop) MK
becomes an indprofinite right RJGKKmodule as in the abstract case So wemay think of minusK as a functor IP (RJGKop)rarr IP (RJGKKop) and consider itsright derived functor
R(minusK) Dminus(IP (RJGKop))rarr Dminus(IP (RJGKKop))
we write HRs (Kminus) for the lsquoclassicalrsquo derived functor given by the composition
LH(IP (RJGKop))rarr Dminus(IP (RJGKop))
R(minusK)minusminusminusminusrarr Dminus(IP (RJGKKop))
LHminuss
minusminusminusminusrarr LH(IP (RJGKKop))
Thus we can compose the two functors HRs (Kminus) and HR
r (GKminus)The case of minusK can be handled similarly
Theorem 713 (LyndonHochschildSerre Spectral Sequence) Suppose K is aprofinite normal subgroup of G Then there are bounded spectral sequences
E2rs = HR
r (GKHRs (KM))rArr HR
r+s(GM)
for all M isin LH(IP (RJGKop)) and
Ers2 = Hr
R(GKHsR(KM))rArr Hr+s
R (GM)
for all M isin RH(PD(RJGK)) both naturally in M In particular these holdfor M isin IP (RJGKop) and M isin PD(RJGK) respectively
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1307
Proof We prove the first statement then Pontryagin duality gives the secondby Lemma 64 By the universal properties of minusK minusGK and minusG it is easy tosee that (minusK)GK = minusG Moreover as for abstract modules minusK is left adjointto the forgetful functor IP (RJGKKop) rarr IP (RJGKop) which sends strictexact sequences to strict exact sequences and hence minusK preserves projectivesSo the result is just an application of the Grothendieck Spectral SequenceTheorem 512
Remark 714 One must be careful in applying this spectral sequence no suchspectral sequence exists in general if we try to define derived functors back tothe original module categories for the reasons discussed in Remark 62 Thenaive definition of homology functor is not sufficiently wellbehaved here
8 Comparison to other cohomologies
Let P (Λ) and D(Λ) be the categories of profinite and discrete Λmodules bothwith continuous homomorphisms We will think of P (Λ) as a full subcategoryof IP (Λ) and D(Λ) as a full subcategory of PD(Λ) We consider alternativedefinitions of ExtnΛ using these categories and show how they compare to ourdefinition Specifically we will compare our definitions to
(i) the classical cohomology of profinite rings using discrete coefficients foundfor instance in [9]
(ii) the theory of cohomology for profinite modules of type FPinfin over profiniterings developed in [12] allowing profinite coefficients
(iii) the continuous cochain cohomology defined as in [13] for all topologicalmodules over topological rings
(iv) the reduced continuous cochain cohomology defined as in [4] for all topological modules over topological rings
Recall from Section 5 that the inclusion Iop PD(Λ) rarr RH(PD(Λ)) has aright adjoint Cop We can give an explicit description of these functors by duality for M isin PD(Λ) Iop(M) = (0 rarr M rarr 0) Each object in RH(PD(Λ))
is isomorphic to a complex M prime = (0rarrM0 fminusrarrM1 rarr 0) in PD(Λ) where M0
is in degree 0 and f is epic and Cop(M prime) = ker(f) Also the functors
RHn D(PD(Λ))rarr RH(PD(Λ))
are given by
RHn(middot middot middotdnminus1
minusminusminusrarrMn dn
minusrarrMn+1 dn+1
minusminusminusrarr middot middot middot ) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
with coker(dnminus1) in degree 0Given M isin P (Λ) N isin D(Λ) to avoid ambiguity we write HomΛ(MN) forthe discrete Rmodule of continuous Λhomomorphisms M rarr N we have
Documenta Mathematica 21 (2016) 1269ndash1312
1308 Marco Boggi and Ged Corob Cook
HomΛ(MN) = HomTΛ(MN) in this case Let P be a projective resolution of
M in P (Λ) and I an injective resolution of N in D(Λ) recall that projectivesin P (Λ) are projective in IP (Λ) and injectives in D(Λ) are injective in PD(Λ)by Lemma 211 In [9] the derived functors of
HomΛ P (Λ)timesD(Λ)rarr D(R)
are defined byExtnΛ(MN) = Hn(HomΛ(PN))
or equivalently by Hn(HomΛ(M I)) where cohomology is taken in D(R)
Proposition 81 IopExtnΛ(MN) = ExtnΛ(MN) as prodiscrete Rmodules
Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi
nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete Rmodules
write dn for the map HomTΛ(Pn N)rarr HomT
Λ(Pn+1 N) In the abelian categoryD(R) applying the Snake Lemma to the diagram
im(dnminus1)
HomTΛ(Pn N)
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn)
shows that
Hn(HomΛ(PN)) = coker(im(dnminus1)rarr ker(dn))
= ker(coker(dnminus1)rarr coim(dn))
Next using once again that D(R) is abelian we have
RHn(HomTΛ(PN)) = (0rarr coker(dnminus1)rarr im(dn)rarr 0)
= (0rarr coker(dnminus1)rarr coim(dn)rarr 0)
so it is enough to show that the map of complexes
0
ker(coker(dnminus1)rarr coim(dn))
0
0
0 coker(dnminus1) coim(dn) 0
is a strict quasiisomorphism or equivalently that its cone
0rarr ker(coker(dnminus1)rarr coim(dn))rarr coker(dnminus1)rarr coim(dn)rarr 0
is strict exact which is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1309
If on the other hand we are given MN isin P (Λ) with M finitely generatedwe avoid ambiguity by writing HomP
Λ (MN) for the profinite Rmodule (withthe compactopen topology) of continuous Λhomomorphisms M rarr N Thenwriting P (Λ)infin for the full subcategory of P (Λ) whose objects are of type FPinfinin [12] the derived functors of
HomPΛ P (Λ)infin times P (Λ)rarr P (R)
are defined byExtPn
Λ (MN) = Hn(HomPΛ(PN))
where P is a projective resolution ofM in P (Λ) such that each Pn is finitely generated and cohomology is taken in P (R) Assume that N is secondcountableso that ExtnΛ(MN) is defined Because P (R) is an abelian category the sameproof as Proposition 81 shows
Proposition 82 Iop ExtPnΛ (MN) = ExtnΛ(MN) as prodiscrete R
modules
For any M isin IP (Λ) and N isin T (Λ) the Rmodule of continuous Λhomomorphisms cHomΛ(MN) with the compactopen topology defines afunctor IP (Λ)timesT (Λ)rarr TAb where TAb is the category of topological abeliangroups and continuous homomorphisms For a projective resolution P of M inIP (Λ) the continuous cochain Ext functors are then defined by
cExtnΛ(MN) = Hn(cHomΛ(PN))
where the cohomology is taken in TAb That is
cExtnΛ(MN) = ker(dn) coim(dnminus1)
where ker(dn) is given the subspace topology and ker(dn) coim(dnminus1) is given
the quotient topology For Λ = ZJGK and M = Z the trivial Λmodule thisdefinition essentially coincides with the continuous cochain cohomology of Gintroduced in [13][Section 2] and the results stated here for Ext functors easilytranslate to the special case of group cohomology which we leave to the readerIndeed it is easy to check that the bar resolution described in [13] gives a
projective resolution of Z in IP (ZJGK) and hence that the cohomology theorydescribed there coincides with oursGiven a short exact sequence
0rarr Ararr B rarr C rarr 0
of topological Λmodules we do not in general get a long exact sequence of cExtfunctors If the short exact sequence is such that the sequence of underlyingmodules of
0rarr cHomΛ(Pn A)rarr cHomΛ(Pn B)rarr cHomΛ(Pn C)rarr 0
Documenta Mathematica 21 (2016) 1269ndash1312
1310 Marco Boggi and Ged Corob Cook
is exact for all Pn then (by forgetting the topology) we do get a sequence
0rarr cExt0Λ(MA)rarr cExt0Λ(MB)rarr cExt0Λ(MC)rarr middot middot middot
which is a long exact sequence of the underlying modulesIn general we cannot expect cExtnΛ(MN) to be a Hausdorff topological groupsince the images of the continuous homomorphisms
dnminus1 cHomΛ(Pnminus1 N)rarr cHomΛ(Pn N)
are not necessarily closed This immediately suggests the following alternativedefinition We define the reduced continuous cochain Ext functors by
rExtnΛ(MN) = ker(dn)coim(dnminus1)
with the quotient topology Clearly rExt coincides with cExt exactly when thesettheoretic image of coim(dnminus1) is closed in ker(dn) Note that even whenwe have a long exact sequence in cExt the passage to rExt need not be exactFor M isin IP (Λ) and N isin PD(Λ) let P be a projective resolution of M inIP (Λ) and (HomT
Λ(PN) d) be the associated cochain complex Then
ExtnΛ(MN) = (0rarr coker(dnminus1)fminusrarr im(dn)rarr 0)
Proposition 83 In this notation
(i) rExtnΛ(MN) = ker(f)
(ii) If dnminus1(HomTΛ(Pnminus1 N)) is closed in HomT
Λ(Pn N) then there holdscExtnΛ(MN) = ker(f)
Proof Consider the diagram
0 im(dnminus1)
HomTΛ(Pn N)
=
coker(dnminus1)
0
0 ker(dn) HomTΛ(Pn N) coim(dn) 0
with the obvious maps The rows are strict exact and the vertical maps areclearly strict so after applying Pontryagin duality Lemma 112 says that
ker(coker(dnminus1)rarr coim(dn)) sim= coker(im(dnminus1)rarr ker(dn)) = rExtnΛ(MN)
On the other hand
ker(coker(dnminus1)rarr coim(dn)) = ker(coker(dnminus1)rarr im(dn))
because coim(dn)rarr im(dn) is monic The second statement is clear
Documenta Mathematica 21 (2016) 1269ndash1312
Continuous Cohomology of Profinite Groups 1311
Corollary 84 cExtnΛ(MN) = ker f for all n in the following two cases
(i) M isin P (Λ) N isin D(Λ)
(ii) MN isin P (Λ) with M of type FPinfin and N secondcountable
Proof In these cases HomTΛ(PN) is in the abelian categories D(Λ) and P (Λ)
respectively so it is strict and the conditions for the proposition are satisfiedfor all n
On the other hand for any M isin IP (Λ) and N isin PD(Λ) we can also considerthe alternative cohomological functors mentioned in Remark 59
LHn RHomTΛ(MN) = (0rarr coim(dnminus1)
gminusrarr ker(dn)rarr 0)
Then we can recover the continuous cochain Ext functors from this information
Proposition 85 (i) cExtnΛ(MN) = coker(g) where the cokernel is takenin T (R)
(ii) rExtnΛ(MN) = coker(g) where the cokernel is taken in PD(R)
Proof (i) In T (R) there holds coker(g) = ker(dn) coim(dnminus1) with thequotient topology which is cExtnΛ(MN) by definition
(ii) Similarly in PD(R) coker(g) = ker(dn)coim(dnminus1)
From another perspective this proposition says that all the Ext functors wehave considered can be obtained from the total derived functor RHomT
Λ(minusminus)Exactly the same approach as this section makes it possible to compare our Torfunctors to other definitions with similar conclusions We leave both definitionsand proofs to the reader noting only the following results For M isin IP (Λ) andN isin IP (Λop) let P be a projective resolution of M in IP (Λ) and (NotimesΛP d)be the associated chain complex Then
TorΛn(NM) = (0rarr coim(dnminus1)hminusrarr ker(dn)rarr 0)
Proposition 86 (i) The continuous chain Tor functor cTorΛn(MN) (defined in the obvious way) is the cokernel coker(h) taken in T (R)
(ii) the reduced continuous chain Tor functor rTorΛn(MN) (defined in theobvious way) is the cokernel coker(h) taken in IP (R)
(iii) cTorΛn(MN) = rTorΛn(MN) if and only if h(coim(dnminus1)) is closed inker(dn)
By Pontryagin duality and Corollary 84 the condition for (iii) is satisfied for alln if M isin P (Λ) N isin P (Λop) or if M isin P (Λ) is of type FPinfin and N isin D(Λop)is discrete and countable
Documenta Mathematica 21 (2016) 1269ndash1312
1312 Marco Boggi and Ged Corob Cook
References
[1] Bourbaki NGeneral Topology Part I Elements of Mathematics AddisonWesley London (1966)
[2] Bourbaki N General Topology Part II Elements of MathematicsAddisonWesley London (1966)
[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216255
[4] Cheeger J Gromov M L2Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189215
[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 12511264
[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)
[7] LaMartin W On the Foundations of kGroup Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)
[8] Prosmans F Algebre Homologique QuasiAbelienne Mem DEA Universite Paris 13 (1995)
[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)
[10] Schneiders JP Quasiabelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)
[11] Strickland N The Category of CGWH Spaces (2009) available athttpneilstricklandstaffshefacukcourseshomotopycgwhpdf
[12] Symonds P Weigel T Cohomology of padic Analytic Groups in NewHorizons in Prop Groups (Prog Math 184) 347408 Birkhauser Boston(2000)
[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257274
[14] Weibel C An Introduction to Homological Algebra (Cambridge Studiesin Advanced Mathematics 38) Cambridge University Press Cambridge(1994)
Marco BoggiMatematica ICEx UFMGAv Antonio Carlos 6627Caixa Postal 702CEP 31270901Belo Horizonte MGBrasilmarcoboggigmailcom
Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom
Documenta Mathematica 21 (2016) 1269ndash1312
 IndProfinite Modules
 ProDiscrete Modules
 Pontryagin Duality
 Tensor Products
 Derived Functors in QuasiAbelian Categories
 Derived Functors in IP(Lambda) and PD(Lambda)
 Homology and cohomology of profinite groups
 Comparison to other cohomologies

Continuous Cohomology of Profinite Groups 1279
(iii) Quotients of M by closed submodules N are prodiscrete with cofinalsequence M(U i +N)
Proof (i) The U i form a basis of open neighbourhoods of 0 in M by [9Exercise 1115] Therefore for any discrete quotient D of M the kernelof the quotient map f M rarr D contains some U i so f factors throughU i
(ii) M is complete and hence N is complete by [1 II Section 34 Proposition 8] It is easy to check that N cap U i is a fundamental system ofneighbourhoods of the identity so N = lim
larrminusN(N capU i) by [1 III Section
73 Proposition 2] Also since M is metrisable by [2 IX Section 31Proposition 4] MN is complete too After checking that (U i +N)N isa fundamental system of neighbourhoods of the identity in MN we getMN = lim
larrminusM(U i + N) by applying [1 III Section 73 Proposition 2]
again
As a result of (i) we call M i a cofinal sequence for M As in IP (Λ) it is clear from Proposition 25 that PD(Λ) is an additive categorywith kernels and cokernelsGiven MN isin PD(Λ) write HomPD
Λ (MN) for the Rmodule of morphismsM rarr N this makes HomPD
Λ (minusminus) into a functor
PD(Λ)op times PD(Λ)rarrMod(R)
in the usual way Note that the indprofinite Zmodules in Remark 19 are alsoprodiscrete Zmodules so the remark also shows that PD(Λ) is not abelian ingeneralAs before we say a morphism f M rarr N in PD(Λ) is strict if coim(f) =im(f) In particular strict epimorphisms are surjections We say that a chaincomplex
middot middot middot rarr LfminusrarrM
gminusrarr N rarr middot middot middot
is strict exact at M if coim(f) = ker(g) We say a chain complex is strict exactif it is strict exact at each M
Remark 26 In general it is not clear whether a map f M rarr N of prodiscrete modules with f(M) closed in N must be strict as is the case forindprofinite modules However we do have the following result
Proposition 27 Let f M rarr N be a morphism in PD(Λ) Suppose thatM (and hence coim(f)) is secondcountable and that the settheoretic imagef(M) is closed in N Then the continuous bijection coim(f) rarr im(f) is anisomorphism in other words f is strict
Proof [6 Chapter 6 Problem R]
As for indprofinite modules we can factorise morphisms in a canonical way
Documenta Mathematica 21 (2016) 1269ndash1312
1280 Marco Boggi and Ged Corob Cook
Corollary 28 (Canonical decomposition of morphisms) Every morphismf M rarr N in IP (Λ) can be uniquely written as the composition of a strictepimorphism a bimorphism and a strict monomorphism Moreover the bimorphism is an isomorphism if and only if the morphism is strict
Remark 29 Suppose we have a short strict exact sequence
0rarr LfminusrarrM
gminusrarr N rarr 0
in PD(Λ) Pick a cofinal sequence M i for M Then as in Proposition 25(ii)L = coim(f) = im(f) = lim
larrminusim(im(f) rarr M i) and similarly for N so we can
write the sequence as a surjective inverse limit of short (strict) exact sequencesof discrete ΛmodulesConversely suppose we have a surjective sequence of short (strict) exact sequences
0rarr Li rarrM i rarr N i rarr 0
of discrete Λmodules Taking limits we get a sequence
0rarr LfminusrarrM
gminusrarr N rarr 0 (lowast)
of prodiscrete Λmodules It is easy to check that im(f) = ker(g) = L =coim(f) and coim(g) = coker(f) = N = im(g) so f and g are strict andhence (lowast) is a short strict exact sequence
Lemma 210 Given MN isin PD(Λ) pick cofinal sequences M i N j respectively Then HomPD
Λ (MN) = limlarrminusj
limminusrarri
HomPDΛ (M i N j) in the category
of Rmodules
Proof Since N = limlarrminusPD(Λ)
N j we have by definition that HomPDΛ (MN) =
limlarrminus
HomPDΛ (MN j) Since the M i are cofinal for M every continuous map
M rarr N j factors through some M i so HomIPΛ (MN j) = lim
minusrarrHomIP
Λ (M i N j)
We call I isin PD(Λ) injective if
0rarr HomPDΛ (N I)rarr HomPD
Λ (M I)rarr HomPDΛ (L I)rarr 0
is an exact sequence of Rmodules whenever
0rarr LrarrM rarr N rarr 0
is strict exact
Lemma 211 Suppose that I is a discrete Λmodule which is injective in thecategory