continuous cohomology and homology of profinite groups · keywords and phrases: continuous...

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 ofpro-discrete and ind-profinite G-modules respectively The standardresults of group (co)homology hold for this theory we prove versionsof the Universal Coefficient Theorem the Lyndon-Hochschild-Serrespectral sequence and Shapirorsquos Lemma

2010 Mathematics Subject Classification Primary 20J06 Secondary20E18 20J05 13J10Keywords and Phrases Continuous cohomology profinite groupsquasi-abelian 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 G-module 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 G-modules does not have enough injectivesWe define the cohomology of a profinite group with coefficients in the categoryof pro-discrete ZJGK-modules PD(ZJGK) This category contains the discrete

Documenta Mathematica 21 (2016) 1269ndash1312

1270 Marco Boggi and Ged Corob Cook

ZJGK-modules and the second-countable profinite ZJGK-modules when G itselfis second-countable this is sufficient for many applicationsPD(ZJGK) is not an abelian category instead it is quasi-abelian 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 t-structure 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 Lyndon-Hochschild-Serre spectral sequence for profinite groups with profinite coeffi-cients This has not been possible in previous formulations of profinite coho-mology 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 cat-egory of coefficient modules to the ind-profinite G-modules 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 Symonds-Weigel theoryfor profinite modules of type FPinfin with profinite coefficients We also definea continuous cochain cohomology constructed by considering only the contin-uous G-maps from the standard bar resolution of a topological group G to atopological G-module M with the compact-open topology and taking its co-homology 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 R-modules M without a topologysuch that giving M the discrete topology creates a topological group on whichthe R-action R times M rarr M is not continuous Therefore a discrete modulewill mean one for which the R-action is continuous and we will call algebraicobjects without a topology abstract

1 Ind-Profinite Modules

We say a topological space X is ind-profinite 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 ind-profinite spaces

and continuous maps


Continuous Cohomology of Profinite Groups 1271

Proposition 11 Given an ind-profinite 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 ind-profinite 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 ind-profinite spaces are compactlygenerated Indeed a subspace U of an ind-profinite 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 con-struct X ⊔ Y using the cofinal sequence Xi ⊔ Yi However it is not clearwhether X times Y with the product topology is ind-profinite Instead thanks toProposition 11 the ind-profinite 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

k-ification 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 ind-profinite topology is an ind-profinite 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 ind-profinite Moreover Qp is ind-profinite via thecofinal sequence

Zpmiddotpminusrarr Zp

middotpminusrarr middot middot middot (lowast)

Remark 13 It is not obvious that ind-profinite abelian groups are topologicalgroups In fact we see below that they are But it is much easier to seethat they are k-groups 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

k-group intuition will often be more useful



In the terminology of [5] the ind-profinite abelian groups are just the abelianweakly profinite groups We recall some of the basic results of [5]

Proposition 14 Suppose M is an ind-profinite abelian group with cofinalsequence Mi

(i) Any compact subspace of M is contained in some Mi

(ii) Closed subgroups N of M are ind-profinite with cofinal sequence N capMi

(iii) Quotients of M by closed subgroups N are ind-profinite with cofinal se-quence Mi(N capMi)

(iv) Ind-profinite 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 anind-profinite group a cofinal sequence for M Suppose from now on that R is a commutative profinite ring and Λ is a profiniteR-algebra

Remark 15 We could define ind-profinite rings as colimits of injective se-quences (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 partic-ular it would be nice to use the machinery of ind-profinite rings to study Qpbut the sequence (lowast) making Qp into an ind-profinite abelian group does notmake it into an ind-profinite ring because the maps are not maps of rings

We say that M is a left Λ-k-module if M is a k-group equipped with a continu-ous map ΛtimeskM rarrM A Λ-k-module homomorphism M rarr N is a continuousmap which is a homomorphism of the underlying abstract Λ-modules 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 Λ-moduleWe say that a left Λ-k-module M equipped with an ind-profinite topology isa left ind-profinite Λ-module if there is an injective sequence of profinite sub-modules Mi i isin N which is a cofinal sequence for the underlying space of M So countable discrete Λ-modules are ind-profinite because finitely generateddiscrete Λ-modules are finite and so are profinite Λ-modules In particular Λwith left-multiplication is an ind-profinite Λ-module Note that since profiniteZ-modules are the same as profinite abelian groups ind-profinite Z-modules arethe same as ind-profinite abelian groupsThen we immediately get the following

Corollary 16 Suppose M is an ind-profinite Λ-module with cofinal sequenceMi




(ii) Closed submodules N of M are ind-profinite with cofinal sequence NcapMi

(iii) Quotients of M by closed submodules N are ind-profinite with cofinalsequence Mi(N capMi)

(iv) Ind-profinite Λ-modules are topological Λ-modules

As before we call a sequence Mi of profinite submodules making M into anind-profinite Λ-module a cofinal sequence for M

Lemma 17 Ind-profinite Λ-modules have a fundamental system of neighbour-hoods 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 ind-profinite Λ-modulesand whose morphisms M rarr N are Λ-k-module homomorphisms We willidentify the category of right ind-profinite Λ-modules 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 R-module

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 abstractR-modules and R-module homomorphisms

Proposition 18 IP (Λ) is an additive category with kernels and cokernels

Proof The category is clearly pre-additive the biproduct MoplusN is the biprod-uct 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 count-able direct sum oplusalefsym0

Z2Z with the discrete topology and the countable di-rect product

prodalefsym0

Z2Z with the profinite topology Both are ind-profinite

Z-modules 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



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 bimor-phism 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 set-theoretic image ofM rarr im(f) is dense so the set-theoretic 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 exact-ness 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)



Proof Note that kernels in IP (Λ) are preserved by forgetting the topologyand so are cokernels of strict morphisms by Proposition 110 So by forget-ting the topology and working with abstract Λ-modules 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 set-theoretic 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 ind-profinite Λ-modules 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 Λ-module so

the sequence defines an ind-profinite Λ-module 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)



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 R-modules

(ii) Given X isin IPSpace with a cofinal sequence Xi and N isin IP (Λ) withcofinal sequence Nj write C(XN) for the R-module of continuousmaps X rarr N Then C(XN) = lim


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 Λ-module on Xi The maps Xi rarr Xi+1

induce maps FXi rarr FXi+1 of profinite Λ-modules and hence we get an ind-profinite Λ-module with cofinal sequence FXi Write FX for this modulewhich we will call the free ind-profinite Λ-module 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



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 ind-profinite modules on ind-profinite 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 ind-profinite 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 Λ-modules Indeed Proposition116 shows that free profinite modules are projective in IP (Λ) and therest follows

2 Pro-Discrete Modules

Write PD(Λ) for the category of left pro-discrete Λ-modules the objects M inthis category are countable inverse limits as topological Λ-modules of discreteΛ-modules M i i isin N the morphisms are continuous Λ-module homomor-phisms So discrete torsion Λ-modules are pro-discrete and so are second-countable profinite Λ-modules by [9 Proposition 261 Lemma 511] and inparticular Λ with left-multiplication is a pro-discrete Λ-module if Λ is second-countable Moreover Qp is a pro-discrete Z-module via the sequence

middot middot middotmiddotpminusrarr QpZp

middotpminusrarr QpZp

We will identify the category of right pro-discrete Λ-modules with PD(Λop) inthe usual way

Lemma 21 Pro-discrete Λ-modules are first-countable

Proof We can construct M = limlarrminus

M i as a closed subspace ofprod

M i Each M i

is first-countable because it is discrete and first-countability is closed undercountable products and subspaces



Remarks 22 (i) This shows that Λ itself can be regarded as a pro-discreteΛ-module if and only if it is first-countable if and only if it is second-countable 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 second-countable

(ii) Since first-countable spaces are always compactly generated by [11Proposition 16] pro-discrete Λ-modules are compactly generated as topo-logical spaces In fact more is true Given a pro-discrete Λ-module 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 dis-crete torsion N i is finite Hence N = lim

larrminusN i is a profinite Λ-submodule

of M containing X So pro-discrete 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 Pro-discrete Λ-modules are metrisable and complete

Proof [2 IX Section 31 Proposition 1] and the corollary to [1 II Section35 Proposition 10]

In general pro-discrete Λ-modules need not be second-countable because forexample PD(Z) contains uncountable discrete abelian groups However wehave the following result

Lemma 24 Suppose a Λ-module M has a topology which makes it pro-discreteand ind-profinite (as a Λ-module) Then M is second-countable and locallycompact

Proof As an ind-profinite Λ-module take a cofinal sequence of profinite sub-modules 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 count-able 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 second-countable because second-countability 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 pro-discrete Λ-module 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 pro-discrete with a cofinal sequenceN(N cap U i)



(iii) Quotients of M by closed submodules N are pro-discrete 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 Propo-sition 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 R-module 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 ind-profinite Z-modules in Remark 19 are alsopro-discrete Z-modules 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



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 pro-discrete modules with f(M) closed in N must be strict as is the case forind-profinite 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 second-countable and that the set-theoretic 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 ind-profinite modules we can factorise morphisms in a canonical way



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 bimor-phism 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 Λ-modulesConversely suppose we have a surjective sequence of short (strict) exact se-quences

0rarr Li rarrM i rarr N i rarr 0

of discrete Λ-modules Taking limits we get a sequence

0rarr LfminusrarrM

gminusrarr N rarr 0 (lowast)

of pro-discrete Λ-modules 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 re-spectively Then HomPD

Λ (MN) = limlarrminusj

limminusrarri

HomPDΛ (M i N j) in the category

of R-modules

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 R-modules whenever


is strict exact

Lemma 211 Suppose that I is a discrete Λ-module which is injective in thecategory of discrete Λ-modules Then I is injective in PD(Λ)



Proof We know HomPDΛ (minus I) is exact on discrete Λ-modules Remark 29

shows that we can write short strict exact sequences of pro-discrete Λ-modulesas surjective inverse limits of short exact sequences of discrete modules inPD(Λ) and then by injectivity applying HomPD

Λ (minus I) gives a direct sys-tem of short exact sequences of R-modules the exactness of such direct limitsis well-known

In particular we get that QZ with the discrete topology is injective in PD(Z)

ndash it is injective among discrete Z-modules (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 Λ-action ie such that λf(m) =f(λm) for all λ isin Λm isin M Consider the category T (Λ) of topological Λ-modules and continuous Λ-module 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 R-module of morphisms

M rarr N in T (Λ) For the following lemma this will denote an abstract R-module after which we will define a topology on HomT

Λ(MN) making it intoa topological R-module

Lemma 212 As abstract R-modules HomTΛ(MN) = lim

larrminusijHomT

Λ(Mi Nj)

We may give each HomTΛ(Mi N

j) the discrete topology which is also thecompact-open topology in this case Then we make lim

larrminusHomT

Λ(Mi Nj) into

a topological R-module by giving it the limit topology giving HomTΛ(MN)

this topology therefore makes it into a pro-discrete R-module From now onHomT

Λ(MN) will be understood to have this topology The topology thusconstructed is well-defined 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 con-struction 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 rightΛ-modules behaves in the same way we may express this by treating MN asleft Λop-modules 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



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

pro-discrete R-modules becauseM andN are bounded below so HomTΛ(MN)

is a complex in PD(R)

Suppose ΘΦ are profinite R-algebras Then let PD(ΘminusΦ) be the category ofpro-discrete ΘminusΦ-bimodules and continuous ΘminusΦ-homomorphisms If M isan ind-profinite Λ minusΘ-bimodule and N is a pro-discrete Λminus Φ-bimodule onecan make HomT

Λ(MN) into a pro-discrete ΘminusΦ-bimodule 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 Λ-module which is injective in PD(Λ)Then HomT

Λ(minus I) sends short strict exact sequences of ind-profinite Λ-modulesto short strict exact sequences of pro-discrete R-modules

Proof Proposition 110 shows that we can write short strict exact sequencesof ind-profinite Λ-modules 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 discreteR-modules the inverse limit of these is strict exact by Remark 29

In particular this applies when I = QZ with the discrete topology as a

Z-module

Consider QZ with the discrete topology as an ind-profinite abelian groupGiven M isin IP (Λ) with a cofinal sequence Mi we can think of M as anind-profinite abelian group by forgetting the Λ-action then Mi becomes acofinal sequence of profinite abelian groups for M Now apply HomT

Z(minusQZ)

to get a pro-discrete abelian group We can endow each HomTZ(MiQZ) with

the structure of a right Λ-module such that the Λ-action is continuous by[9 p165] Taking inverse limits we can therefore make HomT

Z(MQZ) into

a pro-discrete right Λ-module 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 pro-discrete abelian group by forgetting the Λ-actionand then M i is a cofinal sequence of discrete abelian groups Recall that

as (abstract) Z-modules HomPDZ

(MQZ) sim= limminusrarri

HomPDZ

(M iQZ) We can

endow each HomPDZ

(M iQZ) with the structure of a profinite right Λ-module



by [9 p165] Taking direct limits we then make HomPDZ

(MQZ) into an ind-profinite right Λ-module 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 se-quences 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 iso-morphic 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 Λ-modules 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 defi-nitions 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 injec-tive I and a strict monomorphism M rarr I A discrete Λ-module I isinjective in PD(Λ) if and only if it is injective in the category of discreteΛ-modules



(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 Λ-module which is projective in IP (Λ) Then the func-tor HomT

Λ(Pminus) sends strict exact sequences of pro-discrete Λ-modules tostrict exact sequences of pro-discrete R-modules

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) = Qp

The topology defined on Mlowast = HomTZ(MQZ) when M is an ind-profinite

Λ-module coincides with the compact-open topology because the (discrete)topology on each HomT

Z(MiQZ) is the compact-open topology and every

compact subspace of M is contained in some Mi by Proposition 11 Sim-ilarly for a pro-discrete Λ-module N every compact subspace of N is con-tained in some profinite submodule L by Remark 22(ii) and so the compact-open 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) compact-open topology

Proposition 37 The compact-open topology on HomPDZ

(NQZ) coincideswith the topology defined on Nlowast

Proof By the preceding remarks HomPDZ

(NQZ) with the compact-opentopology 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 dis-crete 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 ind-profinite Λ-modules is complete Haus-dorff and totally disconnected

Proof By Lemma 17 we just need to show the topology is complete Proposi-tion 37 shows that ind-profinite Λ-modules 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 ind-profinite Λ-modules 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



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 R-modulesThen 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 isomor-phism 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 nec-essary 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 Z-module which is injective in PD(Z) is divisible On the

other hand the discrete Z-modules 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 Z-modules are injective in the

category of discrete Z-modules and hence injective in PD(Z) too by Corollary35(iii) So duality gives

Corollary 313 (i) A discrete Z-module is injective in PD(Z) if and onlyif it is divisible



(ii) A profinite Z-module is projective in IP (Z) if and only if it is torsion-free

Proof Being divisible and being torsion-free are Pontryagin dual by [9 Theo-rem 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 torsion-free)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 torsion-free 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 quasi-abelian 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 quasi-abeliancategory if it satisfies the following two conditions

(QA) in any pull-back square

Aprimef prime

Bprime

Af

B

if f is strict epic then so is f prime

(QAlowast) in any push-out square

Af

B

Aprimef prime

Bprime

if f is strict monic then so is f prime



IP (Λ) satisfies axiom (QA) because forgetting the topology preserves pull-backs and Mod(Λ) satisfies (QA) so pull-backs of surjections are surjectionsRecall by Remark 22(ii) that pro-discrete modules are compactly generatedhence PD(Λ) satisfies (QA) by [11 Proposition 236] since the forgetful functorto topological spaces preserves pull-backs Then both categories satisfy axiom(QAlowast) by duality and we have

Proposition 315 IP (Λ) and PD(Λ) are quasi-abelian categories

Moreover note that the definition of a strict morphism in a quasi-abelian cat-egory 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 ind-profinite 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 ind-profinite R-module on LtimeskM From the bilinearity of b we getthat the R-submodule 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



(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 Λop-actions 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 prop-erty 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


in IP (Λop) and

middot middot middotdprime

1minusrarrM1dprime

0minusrarrM0

dprime

minus1


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 ind-profinite R-modulesbecause M and N are bounded below so LotimesΛM is a complex in IP (R)

Suppose from now on that ΘΦΨ are profinite R-algebras Then let IP (ΘminusΦ) be the category of ind-profinite Θ minus Φ-bimodules and Θ minus Φ-k-bimodulehomomorphisms We leave the details to the reader after noting that an ind-profinite R-module N with a left Θ-action and a right Φ-action which arecontinuous on profinite submodules is an ind-profinite Θ minus Φ-bimodule sincewe can replace a cofinal sequence Ni of profinite R-modules with a cofinalsequence ΘmiddotNi middotΦ of profinite ΘminusΦ-bimodules If L is an ind-profinite ΘminusΛ-bimodule and M is an ind-profinite ΛminusΦ-bimodule one can make LotimesΛM intoan ind-profinite Θminus Φ-bimodule in the same way as in the abstract case



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 Ψ-bimodules 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 co-covariant bifunctor IP (Λ)op times

PD(Λ)rarr PD(R)) commutes with limits in both variables and that otimesΛ com-mutes with colimits in both variables by [14 Theorem 2610]

If L isin IP (Θ minus Φ) Pontryagin duality gives Llowast the structure of a pro-discreteΦminusΘ-bimodule and similarly with ind-profinite and pro-discrete 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)



5 Derived Functors in Quasi-Abelian Categories

We give a brief sketch of the machinery needed to derive functors in quasi-abelian categories See [8] and [10] for detailsFirst a notational convention in a chain complex (A d) in a quasi-abeliancategory 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 quasi-abelian 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 quasi-isomorphism if its cone is in N(E)Deriving functors in quasi-abelian categories uses the machinery of t-structuresThis can be thought of as giving a well-behaved cohomology functor to a tri-angulated category For more detail on t-structures see [8 Section 13]Given a triangulated category T with translation functor T a t-structure 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 t-structure 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 t-structure 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 t-structure on a triangulated category is anabelian category

There are two canonical t-structures on D(E) the left t-structure and the rightt-structure and correspondingly a left heart LH(E) and a right heart RH(E)The t-structures and hearts are dual to each other in the sense that there is



a natural isomorphism between LH(E) and RH(Eop) (one can check that Eop

is quasi-abelian) so we can restrict investigation to LH(E) without loss ofgeneralityExplicitly the left t-structure on D(E) is given by taking T le0 to be the com-plexes which are strict exact in all positive degrees and T ge0 to be the com-plexes 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 iden-tifying E with its image under I we can think of E as a reflective subcategoryof LH(E) Moreover given a sequence


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 leftt-structure of D(E) with the standard t-structure 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



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 monomor-phism 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 Sup-pose we are given an additive functor F E rarr E prime between quasi-abelian cate-gories 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 quasi-abeliancategories and suppose E has enough injectives

Proposition 57 For E isin K+(E) there is an I isin K+(E) and a strict quasi-isomorphism 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



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 co-homological functor LHn RF The reason for using RHn RF is Proposition55 Indeed when RH(E) has enough injectives we may construct Cartan-Eilenberg 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 bifunc-tors 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 res-olutions 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 quasi-abelian category E has enoughprojectives then an additive functor F from E to another quasi-abelian 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 non-zeroterms 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 quasi-abeliancategories 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))



we can take a bounded below Cartan-Eilenberg resolution I of A Then we canapply Proposition 510 to the bounded below double complex F (I) to get thefollowing result


IEpq2 = RHp(RqF (A))

IIEpq2 = (RpF )(RHq(A))

rArr Rp+qF (A)

naturally in A


Suppose now that we are given additive functors G E rarr E prime F E prime rarrE primeprime between quasi-abelian 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 quasi-abelian this induces a triangulated functor

K+(F ) K+(E) timesK+(E prime)rarr K+(E primeprime)



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 quasi-isomorphismsto 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 quasi-isomorphism A rarr Aprime in K+(E) consider the map of dou-ble complexes K+(F )(A I) rarr K+(F )(Aprime I) and apply Proposition 510to show that this map induces a quasi-isomorphism 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 cat-egories 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 sim-ilarly for the maps We will also identify Dminus(IP (Λ)) with the category of



chain complexes A (localised over the strict quasi-isomorphisms) which are0 in negative degrees by setting An = Aminusn The Pontryagin duality ex-tends 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



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 ho-mology 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 well-behaved homology functor fromchain complexes in a quasi-abelian 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



(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 res-olution 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 apply-ing Pontryagin duality

We will now see that for module-theoretic 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 well-behaved



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



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 ind-profinite right G-modules IP (Gop) to have as its objects ind-profinite abelian groups M with acontinuous map M timesk GrarrM and as its morphisms continuous group homo-morphisms which are compatible with the G-action We define the category ofpro-discrete G-modules PD(G) to have as its objects prodiscrete Z-modulesM with a continuous map GtimesM rarrM and as its morphisms continuous grouphomomorphisms which are compatible with the G-actionFrom now on R will denote a commutative profinite ring

Proposition 71 (i) IP (Gop) and IP (ZJGKop) are equivalent

(ii) An ind-profinite right RJGK-module is the same as an ind-profinite R-module 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 pro-discrete RJGK-module is the same as a pro-discrete R-module 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 anind-profinite abelian group Replacing each Mi with M prime

i = Mi middot G ifnecessary we have a cofinal sequence for M consisting of profinite rightG-modules By [9 Proposition 536(c)] each M prime

i canonically has the

structure of a profinite right ZJGK-module 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

Z-structure on the profinite elements in the sequence These functors areclearly inverse to each other



(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 G-action 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 G-action 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 ind-profinite RJGK-modulesBy (iv) of Proposition 71 given M isin PD(R) we can think of M as an objectin PD(RJGK) with trivial G-action 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 G-actionand 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 pro-discreteRJGK-modules

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



(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 R-modulestructure and think of M as an object of LH(IP (ZJGKop)) that is the under-lying complex of abelian k-groups 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 G-action Then there are non-canonically split short strict exact se-quences

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



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 G-action M =

HomTZ(ZM) where we think of Z as an ind-profinite Zminus ZJGK-bimodule with

trivial G-action 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 torsion-free 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 ind-profinite RJHK minus RJGK-bimodule the left H-action is given by leftmultiplication byH onG and the rightG-action is given by right multiplicationby G on G We will denote this bimodule by RJHցGւGKIf M isin IP (RJGK) we can restrict the G-action to an H-action Moreovermaps of G-modules which are compatible with the G-action are compatiblewith the H-action So restriction gives a functor

ResGH IP (RJGK)rarr IP (RJHK)

ResGH can equivalently be defined by the functor RJHցGւGKotimesRJGKminus Simi-larly we can define a restriction functor

ResGH PD(RJGKop)rarr PD(RJHKop)

by HomTRJGK(RJHցGւGKminus)



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 ind-profinite modules because tensor products do and ResGHand CoindGH commute with limits of pro-discrete modules because Hom doesin the second variableWe can similarly define restriction on right ind-profinite or left pro-discreteRJGK-modules induction on left ind-profinite RJGK-modules and coinductionon right pro-discrete RJGK-modules using RJGցGւHK Details are left to thereaderSuppose an abelian group M has a left H-action together with a topology thatmakes it into both an ind-profinite H-module and a pro-discrete H-moduleFor example this is the case if M is second-countable 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) natu-rally 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



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) projec-tive 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



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 resolu-tion 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 G-action ndash the restric-tion 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 ind-profinite right RJGKK-module 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 (Lyndon-Hochschild-Serre 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



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 well-behaved here

8 Comparison to other cohomologies

Let P (Λ) and D(Λ) be the categories of profinite and discrete Λ-modules 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 topo-logical 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 du-ality 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 R-module of continuous Λ-homomorphisms M rarr N we have



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 pro-discrete R-modules

Proof We have ExtnΛ(MN) = RHn(HomTΛ(PN)) Because each Pn is profi-

nite HomTΛ(PN) = HomΛ(PN) is a cochain complex of discrete R-modules

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 quasi-isomorphism 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



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 R-module (withthe compact-open topology) of continuous Λ-homomorphisms 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 gen-erated and cohomology is taken in P (R) Assume that N is second-countableso 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 pro-discrete R-

modules

For any M isin IP (Λ) and N isin T (Λ) the R-module of continuous Λ-homomorphisms cHomΛ(MN) with the compact-open 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 Λ-module 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 Λ-modules 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



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 theset-theoretic 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



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 second-countable

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) (de-fined 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



References

[1] Bourbaki NGeneral Topology Part I Elements of Mathematics Addison-Wesley London (1966)

[2] Bourbaki N General Topology Part II Elements of MathematicsAddison-Wesley London (1966)

[3] Corob Cook GOn Profinite Groups of Type FPinfin Adv Math 294 (2016)216-255

[4] Cheeger J Gromov M L2-Cohomology and Group Cohomology TopologyVol 25 No 2 (1986) 189-215

[5] Evans D Hewitt P Continuous Cohomology of Permutation Groups onProfinite Modules Comm Algebra 34 (2006) 1251-1264

[6] Kelley J General Topology (Graduate texts in Mathematics 27)Springer Berlin (1975)

[7] LaMartin W On the Foundations of k-Group Theory (DissertationesMathematicae 146) Panstwowe Wydawn Naukowe Warsaw (1977)

[8] Prosmans F Algebre Homologique Quasi-Abelienne Mem DEA Univer-site Paris 13 (1995)

[9] Ribes L Zalesskii P Profinite Groups (Ergebnisse der Mathematik undihrer Grenzgebiete Folge 3 40) Springer Berlin (2000)

[10] Schneiders J-P Quasi-abelian Categories and Sheaves Mem Soc MathFr 2 76 (1999)

[11] Strickland N The Category of CGWH Spaces (2009) available athttpneil-stricklandstaffshefacukcourseshomotopycgwhpdf

[12] Symonds P Weigel T Cohomology of p-adic Analytic Groups in NewHorizons in Pro-p Groups (Prog Math 184) 347-408 Birkhauser Boston(2000)

[13] Tate J Relations between K2 and Galois Cohomology Invent Math 36(1976) 257-274

[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 31270-901Belo Horizonte MGBrasilmarcoboggigmailcom

Ged Corob CookBuilding 54Mathematical SciencesUniversity of SouthamptonHighfieldSouthampton SO17 1BJUKgcorobcookgmailcom


Ind-Profinite Modules

Pro-Discrete Modules

Pontryagin Duality

Tensor Products

Derived Functors in Quasi-Abelian Categories

Derived Functors in IP(Lambda) and PD(Lambda)

Homology and cohomology of profinite groups

Comparison to other cohomologies


ZJGK-modules and the second-countable profinite ZJGK-modules when G itselfis second-countable this is sufficient for many applicationsPD(ZJGK) is not an abelian category instead it is quasi-abelian 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 t-structure 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 Lyndon-Hochschild-Serre spectral sequence for profinite groups with profinite coeffi-cients This has not been possible in previous formulations of profinite coho-mology 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 cat-egory of coefficient modules to the ind-profinite G-modules 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 Symonds-Weigel theoryfor profinite modules of type FPinfin with profinite coefficients We also definea continuous cochain cohomology constructed by considering only the contin-uous G-maps from the standard bar resolution of a topological group G to atopological G-module M with the compact-open topology and taking its co-homology 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 R-modules M without a topologysuch that giving M the discrete topology creates a topological group on whichthe R-action R times M rarr M is not continuous Therefore a discrete modulewill mean one for which the R-action is continuous and we will call algebraicobjects without a topology abstract

1 Ind-Profinite Modules

We say a topological space X is ind-profinite 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 ind-profinite spaces

and continuous maps


















































prodalefsym0



















L

f

Mp

g

N

h

0




















0rarroplus

Ln rarroplus

Mn rarroplus

Nn rarr 0


coim(oplus


fn)) =oplus

coker(ker(fn))

=oplus


fn)) = im(oplus

fn)










C(Xi Nj)


Mi we have that

HomIPΛ (MN) = lim

larrminusHomIP

Λ (Mi N)



minusrarrHomIP

Λ (Mi Nj)

(ii) Similarly







HomIPΛ (FXN) = lim

larrminusi

limminusrarrj


larrminusi

limminusrarrj

C(Xi Nj) = C(XN)


bifunctors














flowast




oplusFXi More-


⊔Yi











M i Each M i


























again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact









minusrarrT (Λ)Mi






larrminusijHomT

Λ(Mi Nj)



larrminusHomT

Λ(Mi Nj) into















minusrarr N1 d1



















Z-module


Z(minusQZ)



Z(MQZ) into






HomPDZ

(M iQZ) We can

endow each HomPDZ









minusminusrarr













hArr f is strict











Qlowastp = (lim

minusrarr(Zp

middotpminusrarr Zp










limlarrminusT (Λ)

HomPDZ























Piwe get HomT

Λ(Pminus) =prod




from Remark 119






HomTΛ(MN)

f1minusrarrHomT

Λ(NlowastMlowast)

f2minusrarrHomT


f3minusrarrHomT























f Qp rarrprod

N



flowast oplus

N


n

xnpn


prodNQpZp


NQpZp







Aprimef prime

Bprime

Af

B



Af

B

Aprimef prime

Bprime







4 Tensor Products


(i) b(l1 + l2m) = b(l1m) + b(l2m)

(ii) b(lm1 +m2) = b(lm1) + b(lm2)












IP (R)

(LiotimesΛMj)

















in IP (Λop) and


1minusrarrM1dprime

0minusrarrM0

dprime

minus1








Λ(MHomTΘ(LN))










sim= limlarrminus

PD(ΦminusΨ)

HomTΛ(Mj Hom

TΘ(Li Nk))

sim= HomTΛ(MHomT

Θ(LN))























by H0 = τge0τle0







LH0 D(E)rarr LH(E)












































IEpq2 = RHp

hRHqv (A)

IIEpq2 = RHp

vRHqh(A)

rArr RHp+q Tot(A)

naturally in A









rArr Rp+qF (A)

naturally in A





naturally in A


(RF )(RG)(A) = F (G(I)) = R(FG)(A)





On the other hand




























Let













RHn




otimesL


LHminusn
























q))pq



Λ(MN) = ExtnΛ(MN)









lowastM)




LΛM










lowastM) = TorΛop



RHomTΛ(MN) = HomT







RHomTΛ(MN) = HomT



Λ(M I))















= HomTΛ(MN)



0rarroplus

N

Zpfminusrarr

oplus

N




0rarr Qp rarrprod

N

QpZp rarrprod

N

QpZp rarr 0



(i) ExtnZ(QpM

lowast) = 0

(ii) ExtnZ(MQp) = 0






ZpotimesZM fotimes

ZM is given by

oplus

N

Mp rarroplus

N



required












Z(Qp N) is not


















(ii) Similarly


(iv) Similarly






MG =⋂

gisinG






MG = limminusrarr



by Proposition 41




with inverse limits



















Hn(GM) = H Z

n(GM) and




Z(Hn(G Z)M)rarr 0








trivial G-action So

HomTZJGK

(PM) = HomTZJGK

(PHomTZ(ZM))

= HomTZ(Zotimes

ZJGKPM)

= HomTZ(PGM)




















note by






Proof










GH MN) natu-

rally in M and N




HomIPRJGK(Ind

GH MN) = HomIP



= limlarrminusi

limminusrarrj

HomIPRJGK(Ind

GH Mi Nj)

= limlarrminusi

limminusrarrj

HomIPRJHK(MiRes

GH Nj)
















rally in MN


GH N




LRJHK Res

GH N



LRJHKopM











H M Then



= MotimesLRJHK Res

GH N




n (HM) and


R(HMlowast)









LHminuss





E2rs = HR


r+s(GM)


Ers2 = Hr


R (GM)
















are given by



minusrarrMn+1 dn+1















im(dnminus1)

HomTΛ(Pn N)

coker(dnminus1)

0


shows that







0


0

0










are defined byExtPn




modules



























0 im(dnminus1)

HomTΛ(Pn N)

=

coker(dnminus1)

0




On the other hand



























References




















Pontryagin Duality

Tensor Products





















































prodalefsym0



















L

f

Mp

g

N

h

0




















0rarroplus

Ln rarroplus

Mn rarroplus

Nn rarr 0


coim(oplus


fn)) =oplus

coker(ker(fn))

=oplus


fn)) = im(oplus

fn)










C(Xi Nj)


Mi we have that

HomIPΛ (MN) = lim

larrminusHomIP

Λ (Mi N)



minusrarrHomIP

Λ (Mi Nj)

(ii) Similarly







HomIPΛ (FXN) = lim

larrminusi

limminusrarrj


larrminusi

limminusrarrj

C(Xi Nj) = C(XN)


bifunctors














flowast




oplusFXi More-


⊔Yi











M i Each M i


























again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact









minusrarrT (Λ)Mi






larrminusijHomT

Λ(Mi Nj)



larrminusHomT

Λ(Mi Nj) into















minusrarr N1 d1



















Z-module


Z(minusQZ)



Z(MQZ) into






HomPDZ

(M iQZ) We can

endow each HomPDZ









minusminusrarr













hArr f is strict











Qlowastp = (lim

minusrarr(Zp

middotpminusrarr Zp










limlarrminusT (Λ)

HomPDZ























Piwe get HomT

Λ(Pminus) =prod




from Remark 119






HomTΛ(MN)

f1minusrarrHomT

Λ(NlowastMlowast)

f2minusrarrHomT


f3minusrarrHomT























f Qp rarrprod

N



flowast oplus

N


n

xnpn


prodNQpZp


NQpZp







Aprimef prime

Bprime

Af

B



Af

B

Aprimef prime

Bprime







4 Tensor Products


(i) b(l1 + l2m) = b(l1m) + b(l2m)

(ii) b(lm1 +m2) = b(lm1) + b(lm2)












IP (R)

(LiotimesΛMj)

















in IP (Λop) and


1minusrarrM1dprime

0minusrarrM0

dprime

minus1








Λ(MHomTΘ(LN))










sim= limlarrminus

PD(ΦminusΨ)

HomTΛ(Mj Hom

TΘ(Li Nk))

sim= HomTΛ(MHomT

Θ(LN))























by H0 = τge0τle0







LH0 D(E)rarr LH(E)












































IEpq2 = RHp

hRHqv (A)

IIEpq2 = RHp

vRHqh(A)

rArr RHp+q Tot(A)

naturally in A









rArr Rp+qF (A)

naturally in A





naturally in A


(RF )(RG)(A) = F (G(I)) = R(FG)(A)





On the other hand




























Let













RHn




otimesL


LHminusn
























q))pq



Λ(MN) = ExtnΛ(MN)









lowastM)




LΛM










lowastM) = TorΛop



RHomTΛ(MN) = HomT







RHomTΛ(MN) = HomT



Λ(M I))















= HomTΛ(MN)



0rarroplus

N

Zpfminusrarr

oplus

N




0rarr Qp rarrprod

N

QpZp rarrprod

N

QpZp rarr 0



(i) ExtnZ(QpM

lowast) = 0

(ii) ExtnZ(MQp) = 0






ZpotimesZM fotimes

ZM is given by

oplus

N

Mp rarroplus

N



required












Z(Qp N) is not


















(ii) Similarly


(iv) Similarly






MG =⋂

gisinG






MG = limminusrarr



by Proposition 41




with inverse limits



















Hn(GM) = H Z

n(GM) and




Z(Hn(G Z)M)rarr 0








trivial G-action So

HomTZJGK

(PM) = HomTZJGK

(PHomTZ(ZM))

= HomTZ(Zotimes

ZJGKPM)

= HomTZ(PGM)




















note by






Proof










GH MN) natu-

rally in M and N




HomIPRJGK(Ind

GH MN) = HomIP



= limlarrminusi

limminusrarrj

HomIPRJGK(Ind

GH Mi Nj)

= limlarrminusi

limminusrarrj

HomIPRJHK(MiRes

GH Nj)
















rally in MN


GH N




LRJHK Res

GH N



LRJHKopM











H M Then



= MotimesLRJHK Res

GH N




n (HM) and


R(HMlowast)









LHminuss





E2rs = HR


r+s(GM)


Ers2 = Hr


R (GM)
















are given by



minusrarrMn+1 dn+1















im(dnminus1)

HomTΛ(Pn N)

coker(dnminus1)

0


shows that







0


0

0










are defined byExtPn




modules



























0 im(dnminus1)

HomTΛ(Pn N)

=

coker(dnminus1)

0




On the other hand



























References




















Pontryagin Duality

Tensor Products

















L

f

Mp

g

N

h

0




















0rarroplus

Ln rarroplus

Mn rarroplus

Nn rarr 0


coim(oplus


fn)) =oplus

coker(ker(fn))

=oplus


fn)) = im(oplus

fn)










C(Xi Nj)


Mi we have that

HomIPΛ (MN) = lim

larrminusHomIP

Λ (Mi N)



minusrarrHomIP

Λ (Mi Nj)

(ii) Similarly







HomIPΛ (FXN) = lim

larrminusi

limminusrarrj


larrminusi

limminusrarrj

C(Xi Nj) = C(XN)


bifunctors














flowast




oplusFXi More-


⊔Yi











M i Each M i


























again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact









minusrarrT (Λ)Mi






larrminusijHomT

Λ(Mi Nj)



larrminusHomT

Λ(Mi Nj) into















minusrarr N1 d1



















Z-module


Z(minusQZ)



Z(MQZ) into






HomPDZ

(M iQZ) We can

endow each HomPDZ









minusminusrarr













hArr f is strict











Qlowastp = (lim

minusrarr(Zp

middotpminusrarr Zp










limlarrminusT (Λ)

HomPDZ























Piwe get HomT

Λ(Pminus) =prod




from Remark 119






HomTΛ(MN)

f1minusrarrHomT

Λ(NlowastMlowast)

f2minusrarrHomT


f3minusrarrHomT























f Qp rarrprod

N



flowast oplus

N


n

xnpn


prodNQpZp


NQpZp







Aprimef prime

Bprime

Af

B



Af

B

Aprimef prime

Bprime







4 Tensor Products


(i) b(l1 + l2m) = b(l1m) + b(l2m)

(ii) b(lm1 +m2) = b(lm1) + b(lm2)












IP (R)

(LiotimesΛMj)

















in IP (Λop) and


1minusrarrM1dprime

0minusrarrM0

dprime

minus1








Λ(MHomTΘ(LN))










sim= limlarrminus

PD(ΦminusΨ)

HomTΛ(Mj Hom

TΘ(Li Nk))

sim= HomTΛ(MHomT

Θ(LN))























by H0 = τge0τle0







LH0 D(E)rarr LH(E)












































IEpq2 = RHp

hRHqv (A)

IIEpq2 = RHp

vRHqh(A)

rArr RHp+q Tot(A)

naturally in A









rArr Rp+qF (A)

naturally in A





naturally in A


(RF )(RG)(A) = F (G(I)) = R(FG)(A)





On the other hand




























Let













RHn




otimesL


LHminusn
























q))pq



Λ(MN) = ExtnΛ(MN)









lowastM)




LΛM










lowastM) = TorΛop



RHomTΛ(MN) = HomT







RHomTΛ(MN) = HomT



Λ(M I))















= HomTΛ(MN)



0rarroplus

N

Zpfminusrarr

oplus

N




0rarr Qp rarrprod

N

QpZp rarrprod

N

QpZp rarr 0



(i) ExtnZ(QpM

lowast) = 0

(ii) ExtnZ(MQp) = 0






ZpotimesZM fotimes

ZM is given by

oplus

N

Mp rarroplus

N



required












Z(Qp N) is not


















(ii) Similarly


(iv) Similarly






MG =⋂

gisinG






MG = limminusrarr



by Proposition 41




with inverse limits



















Hn(GM) = H Z

n(GM) and




Z(Hn(G Z)M)rarr 0








trivial G-action So

HomTZJGK

(PM) = HomTZJGK

(PHomTZ(ZM))

= HomTZ(Zotimes

ZJGKPM)

= HomTZ(PGM)




















note by






Proof










GH MN) natu-

rally in M and N




HomIPRJGK(Ind

GH MN) = HomIP



= limlarrminusi

limminusrarrj

HomIPRJGK(Ind

GH Mi Nj)

= limlarrminusi

limminusrarrj

HomIPRJHK(MiRes

GH Nj)
















rally in MN


GH N




LRJHK Res

GH N



LRJHKopM











H M Then



= MotimesLRJHK Res

GH N




n (HM) and


R(HMlowast)









LHminuss





E2rs = HR


r+s(GM)


Ers2 = Hr


R (GM)
















are given by



minusrarrMn+1 dn+1















im(dnminus1)

HomTΛ(Pn N)

coker(dnminus1)

0


shows that







0


0

0










are defined byExtPn




modules



























0 im(dnminus1)

HomTΛ(Pn N)

=

coker(dnminus1)

0




On the other hand



























References




















Pontryagin Duality

Tensor Products




















0rarroplus

Ln rarroplus

Mn rarroplus

Nn rarr 0


coim(oplus


fn)) =oplus

coker(ker(fn))

=oplus


fn)) = im(oplus

fn)










C(Xi Nj)


Mi we have that

HomIPΛ (MN) = lim

larrminusHomIP

Λ (Mi N)



minusrarrHomIP

Λ (Mi Nj)

(ii) Similarly







HomIPΛ (FXN) = lim

larrminusi

limminusrarrj


larrminusi

limminusrarrj

C(Xi Nj) = C(XN)


bifunctors














flowast




oplusFXi More-


⊔Yi











M i Each M i


























again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact









minusrarrT (Λ)Mi






larrminusijHomT

Λ(Mi Nj)



larrminusHomT

Λ(Mi Nj) into















minusrarr N1 d1



















Z-module


Z(minusQZ)



Z(MQZ) into






HomPDZ

(M iQZ) We can

endow each HomPDZ









minusminusrarr













hArr f is strict











Qlowastp = (lim

minusrarr(Zp

middotpminusrarr Zp










limlarrminusT (Λ)

HomPDZ























Piwe get HomT

Λ(Pminus) =prod




from Remark 119






HomTΛ(MN)

f1minusrarrHomT

Λ(NlowastMlowast)

f2minusrarrHomT


f3minusrarrHomT























f Qp rarrprod

N



flowast oplus

N


n

xnpn


prodNQpZp


NQpZp







Aprimef prime

Bprime

Af

B



Af

B

Aprimef prime

Bprime







4 Tensor Products


(i) b(l1 + l2m) = b(l1m) + b(l2m)

(ii) b(lm1 +m2) = b(lm1) + b(lm2)












IP (R)

(LiotimesΛMj)

















in IP (Λop) and


1minusrarrM1dprime

0minusrarrM0

dprime

minus1








Λ(MHomTΘ(LN))










sim= limlarrminus

PD(ΦminusΨ)

HomTΛ(Mj Hom

TΘ(Li Nk))

sim= HomTΛ(MHomT

Θ(LN))























by H0 = τge0τle0







LH0 D(E)rarr LH(E)












































IEpq2 = RHp

hRHqv (A)

IIEpq2 = RHp

vRHqh(A)

rArr RHp+q Tot(A)

naturally in A









rArr Rp+qF (A)

naturally in A





naturally in A


(RF )(RG)(A) = F (G(I)) = R(FG)(A)





On the other hand




























Let













RHn




otimesL


LHminusn
























q))pq



Λ(MN) = ExtnΛ(MN)









lowastM)




LΛM










lowastM) = TorΛop



RHomTΛ(MN) = HomT







RHomTΛ(MN) = HomT



Λ(M I))















= HomTΛ(MN)



0rarroplus

N

Zpfminusrarr

oplus

N




0rarr Qp rarrprod

N

QpZp rarrprod

N

QpZp rarr 0



(i) ExtnZ(QpM

lowast) = 0

(ii) ExtnZ(MQp) = 0






ZpotimesZM fotimes

ZM is given by

oplus

N

Mp rarroplus

N



required












Z(Qp N) is not


















(ii) Similarly


(iv) Similarly






MG =⋂

gisinG






MG = limminusrarr



by Proposition 41




with inverse limits



















Hn(GM) = H Z

n(GM) and




Z(Hn(G Z)M)rarr 0








trivial G-action So

HomTZJGK

(PM) = HomTZJGK

(PHomTZ(ZM))

= HomTZ(Zotimes

ZJGKPM)

= HomTZ(PGM)




















note by






Proof










GH MN) natu-

rally in M and N




HomIPRJGK(Ind

GH MN) = HomIP



= limlarrminusi

limminusrarrj

HomIPRJGK(Ind

GH Mi Nj)

= limlarrminusi

limminusrarrj

HomIPRJHK(MiRes

GH Nj)
















rally in MN


GH N




LRJHK Res

GH N



LRJHKopM











H M Then



= MotimesLRJHK Res

GH N




n (HM) and


R(HMlowast)









LHminuss





E2rs = HR


r+s(GM)


Ers2 = Hr


R (GM)
















are given by



minusrarrMn+1 dn+1















im(dnminus1)

HomTΛ(Pn N)

coker(dnminus1)

0


shows that







0


0

0










are defined byExtPn




modules



























0 im(dnminus1)

HomTΛ(Pn N)

=

coker(dnminus1)

0




On the other hand



























References




















Pontryagin Duality

Tensor Products













C(Xi Nj)


Mi we have that

HomIPΛ (MN) = lim

larrminusHomIP

Λ (Mi N)



minusrarrHomIP

Λ (Mi Nj)

(ii) Similarly







HomIPΛ (FXN) = lim

larrminusi

limminusrarrj


larrminusi

limminusrarrj

C(Xi Nj) = C(XN)


bifunctors














flowast




oplusFXi More-


⊔Yi











M i Each M i


























again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact









minusrarrT (Λ)Mi






larrminusijHomT

Λ(Mi Nj)



larrminusHomT

Λ(Mi Nj) into















minusrarr N1 d1



















Z-module


Z(minusQZ)



Z(MQZ) into






HomPDZ

(M iQZ) We can

endow each HomPDZ









minusminusrarr













hArr f is strict











Qlowastp = (lim

minusrarr(Zp

middotpminusrarr Zp










limlarrminusT (Λ)

HomPDZ























Piwe get HomT

Λ(Pminus) =prod




from Remark 119






HomTΛ(MN)

f1minusrarrHomT

Λ(NlowastMlowast)

f2minusrarrHomT


f3minusrarrHomT























f Qp rarrprod

N



flowast oplus

N


n

xnpn


prodNQpZp


NQpZp







Aprimef prime

Bprime

Af

B



Af

B

Aprimef prime

Bprime







4 Tensor Products


(i) b(l1 + l2m) = b(l1m) + b(l2m)

(ii) b(lm1 +m2) = b(lm1) + b(lm2)












IP (R)

(LiotimesΛMj)

















in IP (Λop) and


1minusrarrM1dprime

0minusrarrM0

dprime

minus1








Λ(MHomTΘ(LN))










sim= limlarrminus

PD(ΦminusΨ)

HomTΛ(Mj Hom

TΘ(Li Nk))

sim= HomTΛ(MHomT

Θ(LN))























by H0 = τge0τle0







LH0 D(E)rarr LH(E)












































IEpq2 = RHp

hRHqv (A)

IIEpq2 = RHp

vRHqh(A)

rArr RHp+q Tot(A)

naturally in A









rArr Rp+qF (A)

naturally in A





naturally in A


(RF )(RG)(A) = F (G(I)) = R(FG)(A)





On the other hand




























Let













RHn




otimesL


LHminusn
























q))pq



Λ(MN) = ExtnΛ(MN)









lowastM)




LΛM










lowastM) = TorΛop



RHomTΛ(MN) = HomT







RHomTΛ(MN) = HomT



Λ(M I))















= HomTΛ(MN)



0rarroplus

N

Zpfminusrarr

oplus

N




0rarr Qp rarrprod

N

QpZp rarrprod

N

QpZp rarr 0



(i) ExtnZ(QpM

lowast) = 0

(ii) ExtnZ(MQp) = 0






ZpotimesZM fotimes

ZM is given by

oplus

N

Mp rarroplus

N



required












Z(Qp N) is not


















(ii) Similarly


(iv) Similarly






MG =⋂

gisinG






MG = limminusrarr



by Proposition 41




with inverse limits



















Hn(GM) = H Z

n(GM) and




Z(Hn(G Z)M)rarr 0








trivial G-action So

HomTZJGK

(PM) = HomTZJGK

(PHomTZ(ZM))

= HomTZ(Zotimes

ZJGKPM)

= HomTZ(PGM)




















note by






Proof










GH MN) natu-

rally in M and N




HomIPRJGK(Ind

GH MN) = HomIP



= limlarrminusi

limminusrarrj

HomIPRJGK(Ind

GH Mi Nj)

= limlarrminusi

limminusrarrj

HomIPRJHK(MiRes

GH Nj)
















rally in MN


GH N




LRJHK Res

GH N



LRJHKopM











H M Then



= MotimesLRJHK Res

GH N




n (HM) and


R(HMlowast)









LHminuss





E2rs = HR


r+s(GM)


Ers2 = Hr


R (GM)
















are given by



minusrarrMn+1 dn+1















im(dnminus1)

HomTΛ(Pn N)

coker(dnminus1)

0


shows that







0


0

0










are defined byExtPn




modules



























0 im(dnminus1)

HomTΛ(Pn N)

=

coker(dnminus1)

0




On the other hand



























References




















Pontryagin Duality

Tensor Products










flowast




oplusFXi More-


⊔Yi











M i Each M i


























again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact









minusrarrT (Λ)Mi






larrminusijHomT

Λ(Mi Nj)



larrminusHomT

Λ(Mi Nj) into















minusrarr N1 d1



















Z-module


Z(minusQZ)



Z(MQZ) into






HomPDZ

(M iQZ) We can

endow each HomPDZ









minusminusrarr













hArr f is strict











Qlowastp = (lim

minusrarr(Zp

middotpminusrarr Zp










limlarrminusT (Λ)

HomPDZ























Piwe get HomT

Λ(Pminus) =prod




from Remark 119






HomTΛ(MN)

f1minusrarrHomT

Λ(NlowastMlowast)

f2minusrarrHomT


f3minusrarrHomT























f Qp rarrprod

N



flowast oplus

N


n

xnpn


prodNQpZp


NQpZp







Aprimef prime

Bprime

Af

B



Af

B

Aprimef prime

Bprime







4 Tensor Products


(i) b(l1 + l2m) = b(l1m) + b(l2m)

(ii) b(lm1 +m2) = b(lm1) + b(lm2)












IP (R)

(LiotimesΛMj)

















in IP (Λop) and


1minusrarrM1dprime

0minusrarrM0

dprime

minus1








Λ(MHomTΘ(LN))










sim= limlarrminus

PD(ΦminusΨ)

HomTΛ(Mj Hom

TΘ(Li Nk))

sim= HomTΛ(MHomT

Θ(LN))























by H0 = τge0τle0







LH0 D(E)rarr LH(E)












































IEpq2 = RHp

hRHqv (A)

IIEpq2 = RHp

vRHqh(A)

rArr RHp+q Tot(A)

naturally in A









rArr Rp+qF (A)

naturally in A





naturally in A


(RF )(RG)(A) = F (G(I)) = R(FG)(A)





On the other hand




























Let













RHn




otimesL


LHminusn
























q))pq



Λ(MN) = ExtnΛ(MN)









lowastM)




LΛM










lowastM) = TorΛop



RHomTΛ(MN) = HomT







RHomTΛ(MN) = HomT



Λ(M I))















= HomTΛ(MN)



0rarroplus

N

Zpfminusrarr

oplus

N




0rarr Qp rarrprod

N

QpZp rarrprod

N

QpZp rarr 0



(i) ExtnZ(QpM

lowast) = 0

(ii) ExtnZ(MQp) = 0






ZpotimesZM fotimes

ZM is given by

oplus

N

Mp rarroplus

N



required












Z(Qp N) is not


















(ii) Similarly


(iv) Similarly






MG =⋂

gisinG






MG = limminusrarr



by Proposition 41




with inverse limits



















Hn(GM) = H Z

n(GM) and




Z(Hn(G Z)M)rarr 0








trivial G-action So

HomTZJGK

(PM) = HomTZJGK

(PHomTZ(ZM))

= HomTZ(Zotimes

ZJGKPM)

= HomTZ(PGM)




















note by






Proof










GH MN) natu-

rally in M and N




HomIPRJGK(Ind

GH MN) = HomIP



= limlarrminusi

limminusrarrj

HomIPRJGK(Ind

GH Mi Nj)

= limlarrminusi

limminusrarrj

HomIPRJHK(MiRes

GH Nj)
















rally in MN


GH N




LRJHK Res

GH N



LRJHKopM











H M Then



= MotimesLRJHK Res

GH N




n (HM) and


R(HMlowast)









LHminuss





E2rs = HR


r+s(GM)


Ers2 = Hr


R (GM)
















are given by



minusrarrMn+1 dn+1















im(dnminus1)

HomTΛ(Pn N)

coker(dnminus1)

0


shows that







0


0

0










are defined byExtPn




modules



























0 im(dnminus1)

HomTΛ(Pn N)

=

coker(dnminus1)

0




On the other hand



























References




















Pontryagin Duality

Tensor Products




























again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact









minusrarrT (Λ)Mi






larrminusijHomT

Λ(Mi Nj)



larrminusHomT

Λ(Mi Nj) into















minusrarr N1 d1



















Z-module


Z(minusQZ)



Z(MQZ) into






HomPDZ

(M iQZ) We can

endow each HomPDZ









minusminusrarr













hArr f is strict











Qlowastp = (lim

minusrarr(Zp

middotpminusrarr Zp










limlarrminusT (Λ)

HomPDZ























Piwe get HomT

Λ(Pminus) =prod




from Remark 119






HomTΛ(MN)

f1minusrarrHomT

Λ(NlowastMlowast)

f2minusrarrHomT


f3minusrarrHomT























f Qp rarrprod

N



flowast oplus

N


n

xnpn


prodNQpZp


NQpZp







Aprimef prime

Bprime

Af

B



Af

B

Aprimef prime

Bprime







4 Tensor Products


(i) b(l1 + l2m) = b(l1m) + b(l2m)

(ii) b(lm1 +m2) = b(lm1) + b(lm2)












IP (R)

(LiotimesΛMj)

















in IP (Λop) and


1minusrarrM1dprime

0minusrarrM0

dprime

minus1








Λ(MHomTΘ(LN))










sim= limlarrminus

PD(ΦminusΨ)

HomTΛ(Mj Hom

TΘ(Li Nk))

sim= HomTΛ(MHomT

Θ(LN))























by H0 = τge0τle0







LH0 D(E)rarr LH(E)












































IEpq2 = RHp

hRHqv (A)

IIEpq2 = RHp

vRHqh(A)

rArr RHp+q Tot(A)

naturally in A









rArr Rp+qF (A)

naturally in A





naturally in A


(RF )(RG)(A) = F (G(I)) = R(FG)(A)





On the other hand




























Let













RHn




otimesL


LHminusn
























q))pq



Λ(MN) = ExtnΛ(MN)









lowastM)




LΛM










lowastM) = TorΛop



RHomTΛ(MN) = HomT







RHomTΛ(MN) = HomT



Λ(M I))















= HomTΛ(MN)



0rarroplus

N

Zpfminusrarr

oplus

N




0rarr Qp rarrprod

N

QpZp rarrprod

N

QpZp rarr 0



(i) ExtnZ(QpM

lowast) = 0

(ii) ExtnZ(MQp) = 0






ZpotimesZM fotimes

ZM is given by

oplus

N

Mp rarroplus

N



required












Z(Qp N) is not


















(ii) Similarly


(iv) Similarly






MG =⋂

gisinG






MG = limminusrarr



by Proposition 41




with inverse limits



















Hn(GM) = H Z

n(GM) and




Z(Hn(G Z)M)rarr 0








trivial G-action So

HomTZJGK

(PM) = HomTZJGK

(PHomTZ(ZM))

= HomTZ(Zotimes

ZJGKPM)

= HomTZ(PGM)




















note by






Proof










GH MN) natu-

rally in M and N




HomIPRJGK(Ind

GH MN) = HomIP



= limlarrminusi

limminusrarrj

HomIPRJGK(Ind

GH Mi Nj)

= limlarrminusi

limminusrarrj

HomIPRJHK(MiRes

GH Nj)
















rally in MN


GH N




LRJHK Res

GH N



LRJHKopM











H M Then



= MotimesLRJHK Res

GH N




n (HM) and


R(HMlowast)









LHminuss





E2rs = HR


r+s(GM)


Ers2 = Hr


R (GM)
















are given by



minusrarrMn+1 dn+1















im(dnminus1)

HomTΛ(Pn N)

coker(dnminus1)

0


shows that







0


0

0










are defined byExtPn




modules



























0 im(dnminus1)

HomTΛ(Pn N)

=

coker(dnminus1)

0




On the other hand



























References




















Pontryagin Duality

Tensor Products












again

















0rarr LfminusrarrM

gminusrarr N rarr 0






0rarr LfminusrarrM





limminusrarri


of R-modules



limlarrminus



minusrarrHomIP

Λ (M i N j)






is strict exact

Lemma 211 Suppose that I is a discrete Λ-module which is injective in thecategory

continuous cohomology and homology of profinite groups · keywords and phrases: continuous...

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