D E R I V E D - F U N C T O R S OF THE P R O J E C T I V E LIMIT FUNCTO]~

V . K u z * m l n ~ v U 1 ~ 5 1 9 . 4 9

In this paper we sh~U.study the c~unection l ~ e e n the fur,~:~ors re fe red to In the t i t l e and a v ~ l - a=t of re la t ive homoh,g~cal a l g e b z ~

I . Let C A .be the c~tegory of l e f t modules on the r ing A~ 1_et ! ~ a direc~e~ se~; le t CA I be tim c~egory of inverse spect ra of A modu~e~ with directed ~e*. I, and let CI A be the category of dh-eet s~e~ra of A modules with tI~.-.t sxme directed se t L The general r e s e t s of A. G r o ~ e c k [3] on c~1~gory dlzgrams guaz-a.ntees the exis tence of " ~ i c e TM properties in the categories CA I and C l A . In partlcular both these r a re a h e ~ a ~ co~tain sufficiently many In~ectlve and project lve ob- jects' The functor P:CI A ~ CA, which a s s o c i ~ e s with an inverse spec t rum I n project ive (.inverse) limE, is covariant and left exact. The functor P:CIA--'C~ which.~sSOClates with every d l rec t spec t rum its inductive (direct) l lmit , is covari:znt and exact. I n the sequel let p i denote the i th r igl~ der ived f,.mctor of the fv~ctor P:CAL-'CA. An inve r se spect rum ~ is said to be acyclic if p ~ ) == 0 fo r i > 0 .

One of the f i rs t r esu l t s of homolog-Ical a lgebra Is that values of deriv~w] func tor~ can be c o m ~ t e d which decompose objects Inb~ a r b i t r a r y acycl lc resolvents . In tb~ case of the h m c ~ r P.-cAL'~A result is prec.~sely f o r m u l a ~ d as foHow~: le t

be an acyclic r e s o l v e ~ of the spec t rum; then t_be cohomology ~odule HIP(~) of the complex O--P(~|) - - p ~ i ) - - . ~ p ( ~ n ) : - . . . . is canonic~lly ~somorphlc to the, module P ~ ); fu r thermore i f {~] is a n acycl i~ resolvent of the spectrum 7/and if {fi~ is a map of the resolvent {~f~ into the resolvent {r/i~ which coincides with the map f : ~ , on the ~pcc t ru~ ~, then pi(f) ~ HIp(f , ) . We have the following method for construct~ug acyci ie reso lvents . Lot i . - l~T be a monomorphism of the identi ty functor into an exact eovaris~ut functor T whose v~Iues a r e acycl lc . Then the .~or~ul~ d -~ -- 0. ~-t ~ ~ Z ~ ~ ~-~ i Im d ~-~. k : ~ - s ..~ Z~ b~ln~ ~ e ~ncm|c~l ~mO~'p~.__~T~ ~- ~ v'~=~.

~ ~Z, k. z ~ d ~ define inductively the acycl~e resolvent

0 -~ t ~ - ~ ~ . . . . , ~ - ~ . . .

cfthe spectrum ~. For brevi ty we shal l cal l this reso~ven~ a T- re~olven~

A T- re so lven t is na tura l in the ~ense that for any n ~ p f:~--~ it i s e~sy to define by induction the map f , of resolvent~ which coincides on ~ with the map L For the exact sequence 0 ~ - ~ - - 0 th~ sgquence of complexes 0-~P(~ ~*)~P(~z*)~P(~Z*)~0 is exact and ~ e bounded homomorphism~:HIp(~*) - - H i + IP(~l* ) corresponding to this sequence coincides with the connecting homomorphism ~ ' . I ~ $ ) ~ 1M+ ~ 1 ) . By the ~ame token given a T - r e so lven t one can compute the values of the derived func~rm ~n the objects and mappings as well ~s the connecting bomomorphisms. It is evident that there is a

construction for leR reso lvents . Any functor from the category CAinto CA defines canonical ly a funct0r on the category of d iagrams from CA([3]. p. 23). which is denoted in the sequel by the s ame letter as the init ial functor. We slmIJ not pause each t ime this situation ar i se~ but Will s imply a s sume

each functor defined on the category of modules is defined Rlso on the eategory of spec~-~.

2 . Roos [4] indicated a T - r e so lven t for inve r se spect ra . His construct ion consist~ of the following.

for cach inverse spect rum ~ ---- {~=.~=~I o"e considers the spectrum R(~ ={ [I~-~.~S~}- *

s~ose projections coincide with the project ions of the direct product o~ o the factors . If f == { f ~ a ~ I :

v~

Trans la ted f rom 'S~b~r~ldl M ~ t e m a ~ l e h e ~ l Zhurnal . VOLS. ' " �9 . . l o . Z . pp. 333-34s, ~r----------------~Aprfl, I~7. Original a r t ic le submit ted June 3, 1 ~ 5 .

245

exac~ eova "riar.t fmactor R. ~oos es'~abiished that the spect rum P,(~ ) is acyclie for each spectrum ~.

a im raapsi~:~--R(~), defined by (Mr ----- ]1 ~T ffi: ~ - ~ B ( ~ ) . . form a monom0rpPasm of the identtLv functo~ ~rtr~t

Lnto the functor 1~ Consequently tt~ ~eyclio I t - r eso lvem hat been defined. An important property of a t R.resolve~t is that it does not d e p e ~ on tb.~ r ing A. In other wortls if At--A:. is a given x-ing homomoro phL~m and B:CAz---CAt is the cor~---.~ponding functor of riv~ sub.~titution, eR(~)----- Re(~) end 9({') ----- a ( ~ ) " where ~* is an R - r e s o l y e ~ of ~ sI~et~.m~-n ~. I t follows immediate ly t ~ t ~pi(~). = pi0(~) for atry i~ ,er~e spec t rum ~. Another I m ~ t ~roperty of an R-reso lvent is that if the spect rum ~ P.dmits a c o m ~ c t topology then a l l the spec t r a of the R-reso lvent admit a compact topology. Fur the rmore t h e maps dt:~L-~ i§ a r e continuous, t~r,~ a spec t rum of abel ian g r o u p s is said to admit a compact topology if e~ch of the ~:oups does. It is a s sumed that the compact topologies can be chosen s imultaneousiy in a l l these groups so that the projectionB of the spect rum a r e continuous. Froxa this second property of an R-resolvent and from the theore2= of Stcenrod and Eilenlmrg on the ex~ciness of the l im~ sequeno~ of exact sequence~ of compact g r o ~ follows

LEMI~L~ 1, A s p e c t r u m a~m'L~ing a compact topology is acycli~o

With the help of thin l e m m a vae can construct another acycHc T- reso lven t as follows. Let C be the ~,dditive group of r ea l numbers modulo 1; let S(A)be the ftmctor Hom(Hom(A. C)C):Cz--Cz. The functor S.~z -~ Cz is e .~c t and t~e f ~ u l a s ( i t )a(a) (h) m. h(a), a ~ , h~ Hom(~a, C ) def'me a moL,mtor- phism i of the identi ty functor into t ~ ftmctor So The values of the functor S admit a compact topology and ' therefore a r e acycl ic . C o ~ e q ~ t l y a n S-resolvent t~ de,qned in t~e category Cz ! of inverse ~pecttz of abel taa g r o u p ~

3. Yt is known tha'. the v a l . e s of- the ftmctor P do not vary when one passes to the cofinal par t of a spec t rum. -~hen one deletes some project ions of a spec t rmn, when one adjoins new projectism~, o r when one stdmtiiutes for a ' connec ted sec t ion of a spec t rum a moduie in which the projections a r e l so- mor~hisms . We ver i fy that under the~e operations the value~ of the ftmctors pi a l to do not vary . All of these operations a r e convenient ly descr ibed in the following t e rms . Let the map ~-.I~P of directed ~e~ be any m~p o f t b e se ts I ~ P w ~ c h prese rves o rder . F o r every Spectrum { - ~ { a , ~ f l a ~ (direct as well as i nve r se ) w e s e t

If f ----- { f ~ } a ~ P is a l so a map of spectra ,w*(f) = {fro ~)}a~ I is a lso a map of spect ra . This defines the eyact cov~riant functor w*. Any t ~ e a d a = ( ao )a~ ~ in the inve r se spect rum ~ ~ CA P defines a thread j(a) = (ac0(~r))a/~I in the spec t rum r By the same token a functor map J:I~-P~o * is defined. Tim as.~ert!on that the valu e of .the functor P does no t va ry under some s p e ~ r a l t ransformat ions can b e precise ly formulated as follo~,~.

LEMMA 2. Let w :I.--P be a mapping of dlrec~.ed se ts and let the set t0(1) be cofinal in P . T h e n the functor mapping j -.P--Pw ~ is an i somorph~m. .

Proof. F i r s t it. is c l ea r ihat j ~ : P ~ l > w * ({} is a monmorphlsm for any spect rum ~ under the hypothesis that the s e t w(1) is coTL~l in P. We Verify that J~ is �9 an eplmorphlsm. L e t a = (aa)r be a thread of the SlX~ctrum of(~). Fo r every index fl in P we find an index t~ in I such that ~ } = ' f l . We set bfl-~- ~fl~ The e l emen t s bfl do not depend on the choice of the indices cr and form a thread in the ~pectrura ~. i t is c l e a r that j (b) ---- ao The ]emma is p r ~ e d .

l~emark. F o r d i rec t spec t ra the re exists a ftmctor i somorphism j :P~*--P undo" the hypothesis that ~ . I ) is coflnai la I~.

LEMMA 3- If. w(1) is a co f ina l in P the ftmctor w* carr ies~acycl lc spect ra into acyclle spec t ra .

P r o o f . Let # be the functor ~hich associa tes with a module its additive group. We consider the S-resolvent

0-~ I " - ~ Y . . . . - ~ ' - * . . .

of the spect rum e{~}. The resolve~

o - ~ ~.'(~9 -~, ." (~') . . . . . , . ' ( t ' } - ~ - - - of the spect rum t0*{~) i s acycl!c sLice the spec t ra of this resolvent admit a compact topology. In view cf tlds tl]Po*{~ *) = p~t0*0({) ----- 01>i~*{~). B ~ by Lemma 2 the complex pw*(~*)is isomorphic to the

2 4 a

complex ~ * } . Conse~ently e P t ~ ) ~ pte(~)ffi 0pt{~ . I f t t~ spectrum is aeycllc ".hen ~ , - ( ~ ) -=0 and p l ~ * ~ •= 0 (1>'1), that L~ the spectrum ~"~(~) iS also acyclic. ~ne lemma is proved.

THEOREM 1. Le t w-.I~l ~ be a map of dlrec-~d'sets sue~ f~xt oJ(I) is coflnxl Inthe set P. exist unique isomorphismB of reactors jn.~__=;pnop satisfying the following conditions:

Z) for every exact sl:ectz~l sequenc~

O-,-L-~h--~ b-.-O ~ CAr the foIlowing dixgramis commutative:.

"1 "t

P r o o f . Let ~ belor~ to CA P and let ~* be an InJectlve resolven~ of it. By Lemrna 3 the eompleg a,*(~ *) is an acycHc re.solvent of the spectrum t0*(~ ). Consequently Hnpw*(~*)~ pn~0~),

On the other ~ n d the complex I~0"{~*) is natttrally isomorphic to the complex 1~(~ *) since HnF~ *(~*) ---"Im(fL We denote tLis isomorphism by J ~ n : p n ( ~ ) - - ~ ( ~ ) , It is clear tha~ the Leo- morpb~ms j u form a func~or isomorphism jn:pn-.pnr The r emaL~er of the t.~_~r~m L~ proved by standard a_rgume.Z~o

4. In thln section some basic ~oncepts and the0ren~ of relative h0mological algebra are e n u m e r - [zl.

In abellan categories one considers a class Mof monomorphlsms and eplmorphlsms satisfying f o H o ~ g a ~ o m . .

i) If f is an fsomor~l~m f belongs to ~ .

In the exact sequence O--A, iAPA---C the mon~morphiSm I belongs to M ffand only l f t ]~ ephnorphtsm p belongs to M.

3) If f belongs to M, g belongs to M and f and g are monomorphlsms (epimorphisms) then the mo~omorphism f ' g belcegs to M.

4) If f-g b~longs to M and f-g is a monomorphism (eplmorphi~sm) then g be|ongs to M (f beI~ng~

An object F IS said to be M'projecttve if for any epimorphism p:A~B from M and for any bonm- mcrph~m f-.F~-B there exists a homomorphism g:F--A, such that pg ----- g . We say that a category has sufficiently many M-projective objects if for any object A of the cat_~%-or~there e~ists au epimorphi~m F~A~0 of class M with M-projective object F. The left rezolvens o..~_.~An_.... ~A*-. 0 cf ~.n object A is sai d to be an M-resolvent if the objects A n are project ve and the C~ker eplrnorphisms belong t o 11. If a category ~ sufficie~tly many projective objects any object of the category h~s at least o n e ll-resolvent. Any homomorphlsm of A---B which Is one to one to within bomotopy is contInued Ins ~ p of M-resolventso This permits us as in absolute hom01ogic~l algebra to define r~.ght derived functors of contravarlant functors and left derived functors of covariant funct0rs. We denote right derived f t m ~ r s o f a c0nt ra~r iant ftmctor T in the sense of relative homological algebra by MWro

The s e~uence of functors Rir~T) (n --- 0. i . . . . ) form M. a S-.~unctor. tlmt is for any exact s e - quence 0--A,=-A-~A~,~--0 in which i. p belong to M. the natural connecting homomorphlsms B:MnT(A,)-- IIU+LF(A" ) ~ e defined; f u r t hemore the s equen~

. . . ~ u . r ( A ' ) - * M . T ( ~.) -.- m.rfA' )~ M - - ~ T ( A ~ - . .

is exact= The derived functors Horn (,B) are denoted by ~ ( . B ) . Since the functor Hom(,B) i s left exact, ~-(, B) ~ Horn(, B)~ The map cf the projective resolvent of an object A into its M-projectlvo resolvent which eolncides on A with the identity mapplnggenerates the natural mappings pn: ~ A . B) - - Extn(A. B). It turns out that the mapping pt is alway~ a mcnomorphism so One can assume tl~t the ~r0up 0~(A, B) is a Subgroup of the group Extl(A, B). If we consider the elemei,ts of the group Ext'(A, B)

24T

r i n s e e x t e ~ i o ~ in which i, p belong to M.

T h e theory of M- in jec t lve chject.~ is cons t ruc ted d~mrly. 1~ ~t c a t e g o r y has s ,~ f ic ien i ly m m l y M~-projective aucl M-IrJec t lve ob jec t s the r ight de r ived func~or~ of the f u r . ~ T Hom(A,B) de/ ined by tlw decompos i t ion of A Into a r ight r e so lven t and those deTmed by t~e decompos i t ion o f B into �9 L,~R resolve~ coln~da.

5. In this section we con~i~er a special c a s e of the theoz 7 stated above.

Definition., A module monom0rphlsm 0- - -AIB is ca l led se rv ing if for e v e r y submodule Q ' o f the f r e e module Q with a f inite number of gene ra to r s and for eve ry moromorphlsm_ f : Q ' B tz ldng Q' i(A) there exists a mapping f:Q--A such th~ ~ ~-~ f on Q~. An eplmorph~m APc-~O is called ser~ �9 if for every module F with a t-L~ite number o~gen~rators ~.nd for any homom-orphism g:F--C there ex i s t s a homomorph i sm ~ :F-~A such that p-g ----- g,

In t h e sequel l e t M denote the c l a s s of s e rv ing e p lmorph i sms and monomorph i sms .

LEMMA 4. The ClRss M of serving eplmorphisms and monomorphlsms satisfies the class axioms.

Proo f . The ver i f ica t ion of ax ioms I) , 3), and 4) is t r iv ia l . I t i s s l igh t ly l e s s t r i v i a l to v e r i f y ax iom 2 ) . We cov~ider the c o m m ~ a t i v e d i ag ram

o--, Q ' A q ~" F--,.o

J �9 �9 O--... A ' - . , . A - - , A -..0.

w h e r e t h e r o w s a r e e . x l ~ . . . ,

only if ~ne~e exists a homomorph i sm ~'.Q-~A' such that ffl ffi= h.

Ac tua l ly these bomomorph i sms de te rmine e~ch other" accord ing to the . fo rmula ~ + g a ~ ~. The mo~omorp'ni-~m i is servL~g if and only if in Rny dia~ram (1) with ~c~ module Q with ~itr generators there exists a homomorphism f~.ud th~ epimorphism p is serving Lf and only if a homomor- ph i sm g exists in these ~ . Consequently the menom0rphlsm i is s e r v i n g if and only if the

L e t us cons ide r the functor Z.-CA~C A which a s s o c i a t e s with the module A the d i r e c t sum Z(A) o f aH dL~Inct submodules of A with f ini te ly many gene ra to r s ; which ass0ci~f_,~_ with the homomorph i sm f-.A~B the homomorphism ~- {f)-Z (A) - -Z (B); which c a r r i e s the su~rnand A ~ of ihs d i r ec t sum Z{A} into the summand f(Ac0 of the d i r e c t sum :~(B)and which coh-~cides with the mapping f on Ate. The d i r e c t s u m of the ident i ty embeddings Aa- ; -A is t l ~ ep imorph i sm ~'(A)~A. We denote this ep lm0rPhlsm by ~r A. The e p i m o r p h i s m s tr A a r e na tura l and t he r e fo re form a functor epLmorpl~It:m cr : Z ~ I . F ~ " e v e r y module A the module Z(A) i s M-pro jec t ive and the eplmorphLsm u~ is s e r v i n g . Consequent ly the c a t e g o r y C A c o ~ . ~ , ~ suf f ic ien t ly many M-pro jec t i ve module~.

6. The b a s i c aLto o f this s e c ~ o n is tl~e proof of the f o l l o ~ u g a s s e r t k m .

THEOREM 2. Let A be a Noether ian r ing; l e t I be an a r b i t r a r y d l r ec t e~ se t ;~ ~ { ~ , ~ f l a ~ a ~ I be a d i r e c t s p e c t r u m of A-modu les each of which is decomposed into a d i r ec t sum of A-modu le s with f in i te ly many g e n e r a t o r s ; and l e t E be an a r b i t r a r y A-module . Then the g roup PnHom(~, B) i s ~.~t=m-~II~ i somor l~ i~ : to the group on(P(~). B ) .

F i r s t we prove a I ~ n m ~

.LEMI~IA 5. Let A be a Noetheri~n ring,~--{{a,~fla}a6I be an a r b i t r a r y d i r e c t spec t rum of A - modules , and B be an a r b R r a r y A-module , Then t h e r e exists a s p e c t r a l 8equence beginning with t he t e r m EP~q----- pP0q(~, B). and converging to the t e r m E P~,, which is a s soc i a t e d with an app rop r i a t~ image of the f i l t e r e d group ipn(p(~). B) .

P roo f . ~,Ve define the funetor R:CAI--CA I by analogy with the functor P,.-C~K--CIA_ wRh which we. cons t ruc t cd in sec t ion 2 a l e a acyc l I c R- r e so lven t of i nve r se s p e c t r a . L e t P-(~)a -~ ~ " R ( ~ ) f f i

{]~(~c~ ,~)a~I~ f l ~ being embeddlngs ~f ~t summand into a d i r e c t sum. The fo rmu la s "

2 4 8

(l~)~a) ~ ~pa(a) . a belong!ng ~.~, a > p d e , no an eptmorphLsm p: l l~! of the f ~ c t o r R into the IdenIRy ~ o r , T ~ f ~ o r ~z:cr~-c~ ~-~ the ~ , ~ m o ~ , o ~ ep~orp~:~ , ~.p-.e,.- d~n~e ,~ l e e ~ - resolvont in the category C1 ~x of direct ~pe~tra. Let ~ be an P,~*resolv~nt of **he spectrum ~. We cor~ider the complex Eorn(~*, B) of inverse s p e c t r ~

The spectral hyperbomolog~ func'.~rs for P ([3]. p. 47) h~ve lnlLh~1 terms ,EP.zq = ~ ( t * , B), -XP~q== PPBqHom(~**,B) ~nd c o n ~ r g e to t geh~r~ ILr~it. By the abovo mentioned theorem of 1~oos tlm spectra Hom{~ n, B) are acycHc since they r~present the spectra o f direc~ products and the preJectlo,~ in them are the projections of dlrect produ~.s onto the f ~ o r s . .

Consequently the first spectra] s e q u e ~ e is degenerate and converges to the group H-~PHom(~ *, B)=- HnHom(P(~C),B). The complex P(~*) is a r~o lven t of the module v~) where the modules p(~n) are the direct sums of modules with finitely many g=-,~nerators and hence MoproJeCttw~

To establLsh tha'. the differentlzls in ~ complez P(~*) have Serving kernel Rnd cokernel it suffices to ~bow that the epimorpb~s~ !~Jr p,~,~ ~ .- 'o~'~- 'P{~-: , _ . . - . is servLng. Let Q b~ ~n ~rbitrary A.~-ocl'oJ~ w ~ finRely many generators Rnd let g.4~. p(~) be an arbi t rary homomorphtsm~. Since the ring A ta /~>e~herian there exists an index r in I and a t~omomorphL~m g a . ~ - - ~ , such th~ g ----- q a ' g ~ where ~ a Ls the projection o f a module offhe spectrum ~ into the limit module. Since the epimorphisms r ~-~_ p~are serving ther e exists a bomomorphisrn ga -'Q-~RX(~k~ such tb~ ((o" "p)~)a-~= g~. Let ~a: R~(~f-'PR~:(~) be a projection in the spect~u-n RZ(~). We set ~----~ @a-gc~. It is obvious that p{(#. p)~)*~ = g and that the ep~.morph~m P~(r is serving. Consequently the complex P(E*) is am M-resdivent of the module P{~) and the groep_ KnHore..(!~*), B) is isomorphic to the group 9t~l~t~}.B).

second spectral sequence has |,!t!z.l te rm *,EP'~ q =pPHqHorr,{~*. B)) ~ PP~q(~, ~) and converges t~ t.~ group #n(p(~, B). The l e ~ , ' ~ is proved,.

Proof of Tl~eorem 2, If the conditions of the theorem are satisfied the modules ~a a re M-proJec-- tire. But then inthe spectral sequence of ~h~ preceding lemma the terms EP,tq = 0 for pZ I and tim sequence is degenerate. By. the same token ~ ' ~ ~- E ~ ~ ~n(P(~),B)pn Hom(~,B) =~ (P (~ ) ,B ) . TI~ theorem is proved.

I1e~ark.. ~ in the proof of Lcmma 5 o~e r e p ~ . . s t he /~ - reso lven t ~f the spectrum ~ with i ~ projective resolvent the resulting spectral s~ue~ce coincides with the Sequence of Roos [4] ~hich begins with the term EP~q ~-~ PP~tq(~ .B) a~d converges to the term associated with F~ct~P(~),B). The map ~Ja projective re~oivent of a spectrum into an R~-,resolvent of it which coincides on ~ with the identity map defines a map of the sequence of Lemn~ 5 ~to the sI>cctral sequence of Roos. In addRJon on tim t~nn EP~q the map coincides with the homomorphism

in the J|mlt. it coincides with the homomorphism pn: ~(p(~), B)~Extn(p(~), B).

~. In th~ section we show how a map of a spectral sequence can be used to study the interchange- a~Ry of the ~unctor Ext n with the dLrect limit inthe fh-st argument. L e t ~ = { ~ , ~ . ~ 0 f } be a direct spectrum and let ~a :~a- 'P(~) be a project ic~ The limit of the maps Extn(~a. B) transforms Extn(P(~). B) into PExtn(~,B). We denote this map b y ~ . One defines ez~lo~o~sly the homomorph/sm

The map of ~be spectral sequences gives the commutative diagram

Ext'(P (~), B)e*:-p ExP(~,

If a spectrum ~ consists of M-projective modules, in p~rtlcular if It consists of direct sums o~ modules with finitely many generators; then p~n{~, B) = 0 for n > 1. Consequen~ly pn.~n == 0 and tl~. ker~..el of tho homomorphism ~ contAh~s the f.mago of the' bomomorphism pn. In the case n ~ 1 o~B can sharpen th~s result. The exact sequencc-~J of :he.lower st~ps of the spectral sequences form the f~llowLng commutative d l ~ :

0--* p t l l o m (T,. B) --* ~ (P (~..J~) - - , POt (~., ] / ) --~ P , H e m t ~ B) --* 0 t (P(~, B)

a - , p l lom{~ . B~-~ .Extt : (P~) . B} - ~ I ~ F.xtt(~ B)-~Pttlom(7,.. By-,I-~ttt ( P ( ~ . B ~

2 4 9

For th e spectrum ~ of M-pro~ectlve modules this diagram ~ives tim following exact Jequeace:

o -. 0, (P (~,)., ~) ~ ~:~t, (P {~), ~) X P z~t, (~ ~) ~- o~ i,~.(~,), ~)-~ ze., (P (~), e).

~-~_ch indicates the kernel ~ the cokcrnel of the h o m o m o r ~ ~L.

8. Let 0- '~--~ L- . . . . . ~ n - . . . . be an exact sequence of spectra or equl=~Ientlyan inv~rse spcc~ru~.~ of e.~.ac~ sequences. We conslder.a~aln the Spectral hyperhomology functors for P. Tl'm f irst spectral sequence has |niti~.l term ,EP~q --= HPP~ *~ ), and the second spectral s~iuencv has initial te~m '~EP,~q ~ PPHq~ *) ----- 0. Consequently the first spectral sequence converges to zero. This proves.

THEO.I1EM 3. For any exact sequence of.spectra

0-~ y - ~ p - * . . . - ~ ~ - * . . . there exists a s p e ~ ' a l sequence which converges to zero and which begins with the term EP~ q ~-~ H P I ~ * k

"l~is theorem can be need to compute the dimension of the inexactness of the limit sequence of a spectrum of e~act sequence~.

COROLLARY 1. In the exact sequence of spectra

let the spect ra ~ i~ for i-~n satisfy the condition pn-i+~l-~) - - O. Then the sequenee

COROLLARY.2. Let tl~ sequence of spectra 0--~--~--~--~ be exact. Then ~P(~)

COROLLARY 3. In the exact sequeuce of s ~

o -~ ~'-~ r - ~ . . . -* r; - .

l ~ the s v ~ r a ~ be s.~-~ t ~ ~ ( ~ = o for n > 1. Then H~P{~ *) ffi H n - ~ * ) .

9. In this section we examine in detail spectra of a b e l i ~ g-l-ov4~.

The cor~cept o~ a se~.~-~ m~nomorpb_L~m in the cat, go~y of Rbel.inn groups coincides wlth the concept of a rnonomorphism on a servir~ subgroup. Since any subgroup of an abeUan group which la deccndposab]e into a dire~ sum of cyclic groups is iLse!f decomposed into a direct sum of cyclic groul~ mW abelian group has an M-resolvcnt consisting of all of the two specLra. ~.~,~,,~.,.j~ ............. .~'~..-,a- --,~ .---_ n for i -> 2 for any abel.Jan groups A and B. The group 01(A0 B) is the subgroup of serving extensions In the group Ext(A. B) and as Fuchs showed ([6], p. 246) this subgroup coincides with the subgroup of elements of infinite height of the group Ext{A. B), Therefore the following is immediately deducible from Theorem 2.

TIt'EOREM 4. Let ~ be a direct spectrum of groups which are decomposable into a direct sum of cycl ic groups. Then Pittom(~, B) ~--- 0 for i >-- 2 and for any group B and the group P~Hom(~, B) coincides .with the subgroup of elements of infinite height of the group Ext(P(~), B).

The direct sums of cyclic grouvs are M-projective objects and are t.he only M-pro~ectlve objects in the category Of abelian groups. Algebraically compact groups are the M-lnjective objects In tht~ category. If B is an a]geb~ically compact group the spectral sequence of Lemma 5 turns out to be degenerate a~l yields the isomorphism

/ ) I liom ( ~ B ) -~- O " ( P ( ~ , B ) ~ 0 f o r ~ ~ 11_ .

Therefore for any direct spectrum ~ of abel~n groups and for a~y algebraically compact group B the spectrum ITom(~, B) ls:acyclic. In partic~|~r it is acyclic if BIS complete or admRz a compact topology.

Theorem 4 can be used to compute groups pl(~} for a ra ther broad class of spectra o f abeltam

T_HEOREM 5. Let B b e a n arbRrary group.

a) If{ is a direct Spectrum of groups which are decomposable into a sum of cyclic groul~. PiExt{L B) ----- 0 for 1>o.

b) If ~ is an [ n v c ~ spectrum of finite groups

c) If ~ is an invoice Spectrum of groups with finitely many generators, pi([ | B) =- 0 for !>1 and p1(~ @ B)---- Ext(Hom (P(~), Z), B),

d) If ~ is an inverse spectrum of groups with finitely many generators, P ~ * B ) == 0 for | > 1 and pI(~.B) = 0~(Pt(~, B) where t(-~-) is a spectrum iv which the groups e re the dtmls in the semse of PontrL~gin of the maxhn~ periodic subgroups of the groups of the spectrum ~,

Prcof~ a) The term EP'2 q of Roos ' s spectral sequence is eqtml to zero when q-~ 2 for the ~ r y of abelian g ~ u p s . Therefore there exists the exact sequewm

0-~ ~, Horn (~. B)-. E~t (P (D, B) ~ PEn (L B)~ ~" Horn (L ~ - ~ 0

and the isomorphism s Pi+~Hom(~, B) ---'PiExt4~, B) for 1> 0. If the groups of the spectr-mn ~ a re de- composable into a direct sum of cyclic groups, by Theorem 4 PiHom(~, B) -~-- 0 for 1> 1. Consequently. fEzt(~, B) ----- 0 for l> o.

b) Let C be the ad~tive ~roup of r~.al numbers modulo one. For any finite group A the groups A | B and F.xt(Hom(A, C)o B) are naturally isomorphic. Consequently pi(~ ~ B)------ pi Ext(Hom(~, C), B) =ffi �9 0for i - L

c) Let t(~) be the spectrum of maximal periodic subgroups of the spectrum ~ and r(~) ----- ~/t(~). The sequence

is exact. Since the groups t(~)0t are finite by part b) of the theorem the spectrum t(~)~B is acycHc. Consequently pt{~ | B) ---- P~r(~)| B) for i-~ 1. For a group A with a finite number of generating groups r(A)@ B is isomorphic to- the group Hom(Hom(A , Z), B) o Consequently

P~(r(~) | B ~ P~ Horn (Horn (L Z)., B) ~ 0 ~ (Horn (P(~), Z), B). Bence

p,(r~| B) -~- 0 for f > 2 , / ~ ( t | B)--~ E~! (Horn (P(~,). Z), B).

a) P~(~ �9 B ) = P~(t~) �9 B) ---- ~,t eom(t(~i ~ B) . since f o r any nn~te group A the ~ronp~ A* B and Hom(~% B) are naturally isomorphic. Hence" P~ * B) ----" 0 for i> I and P~ �9 B) ----- pJ Horn(t(---D, B) =ffi 0~.T) , T~), _~-e t ~ _ r e ~ '._= ~-r,_~._ ~ _

The. exact sequence of section 7 in the C~tegory of abeltan groups assumes the following form

0,--~ O'(P(~), B) ~--~ Ext(P(~), ~'~

This sequence is exRct for a~y direct spectrum [ of groups which are decomposable .into a direct sum of cyclic groulm.

10. We introduce an example of an inverse spec~-um ~ of abelima groups of which 1~(~), �9 0.

The group G ---- HZ (the direct product of denumerably many infinite cycl!e g roups) has the folio- propert ies:

a} Every ~mbgroup of G with finitely many generators is contained In a direct summand with Finitely many generatorSo

h) The group Hom(G. Z) is isomorphic to a free abelian group with countably many generators. The group Ext(G, Z) ~0 .

Direction by set inclusion of the d i rec t surnmands of B with finitely many Ken, rators follows from tim first property. We denote the spectrum of these direct summands by g. We consider :be iuverse spectrum ,/---- ttom(g, z ) . The projections In this spectrum are epimorphisms and the limR group coincides with the group Horn(G, Z).

Let us consider another inverse spectrum 7.in which the groups are isomorphic to Horn(G, Z) and projections a re identity isomorphisms. The maps ~a :P( :~)~ /a form an epim0rphism ~ of the

Spectrum y onto the spectrum ~L Let ~ be the kerne~ of this eplmorphism. The sequence

251

O-~, {D-~P{v) -./,{n)-+P,{D--~,(-D-~P~T;}-+ ./~{~.j---m{~}.

Is exac'.. It follows.easily from Th'corcm I that the spectrum 7 is acyellc. Th~ map P { ~ ) Isan Iso- morphism, p1(~) __ Ext(G, Z)~ 0 and pi(q) = 0 for i> 1. It follows that pi(~) --- 0 for ! ~ 2 and ~.I(~} =, rx t (TiZ, Z ) ~ O. The grou.ns ~ a of the s p e c t r u m ~ a r e f r ee L,~-oups with countably many g e n e r a t o r s and the p ro jec t ions in this s p e c t r u m a r e m o n o m o r p h i s m s s o this spec t rum can be considerecl a s the spect rum of subgroups of the g roup Z Z with p ro j ec t ion - embedd ings . E v e r y subgroup of the s p e c t r u m ~ is a d i r e c t stu'nmand in ZZ whose c o m p l e m e n t a r y s u m m a n d has f in i te ly many g e n e r a t o r s .

11. ~:~ topology i n v e r s e s p e c t r a a r e used to de f ine s p e c t r a l hor.iology g r o u p s . One c o n s t d s r s the d i r e c t e d ~ �9 ~e, ofall Finite open covers of a space X. The homology group.~ of nerves of these curv~ form the inverse spectrum

h = h . (x ; c) = {# . (x~; 6"), (~ , ' ) . } .

The l imR of tb.js speetru~-r~ ":s by clef'tuition the g r o u p Ha[X-G), that is the s p e c t r a l homology group of the s p a c e X witn coef f ic ien ts f r o m G. We compute the va lues of the functors p i on the s p e c t r u m hn(X;G ). We denote by h n ----- hn(X) the s p e c t r u m o f cohomolog ies with i~nteger coeff ic ient3 of ne rves of f i rdte open cc~,ers of X. The m ~ v e r s a l cc.cff icients fo rmula g ives the following sequence

0 --,- Ext(h "§ G) -~/t= --~ liom (h', G) -4- 0.

Since the groups h n§ 1 have f in i t e ly man. y g e n e r a t o r s t h e s p e c t r u m Ext(h n+l, G) is a e yv l i c by Theorem 5. Hence p i (h n = 0) for i ->2 and Pl (hn) = 01{Hn(X),G). Thus we have proved

THEOREM $. Lett h a be an i n v e r s e s p e c t r u m o f groups whose kimit g roup is b y defL-Ation the group Hn(X;G ). Then P (hn)-~ 0 for i->2 and the gTOUp P~(hn) is isomorpl'Jc to a maximal complete subgroup of the group ~xt(Hn(X),G).

This theorem can easily be used to evaluate the exactness of a sequence of homology groups of a pair {X, Y ) of bicompacta. The sequence of homology group~ of a pair (X, Y) Is by definition the lirnR scquence of the following sequence of inverse spectra=

. . . ~ h . { Y : C ) ~ h . ( X ; C ) ~ h . ( X . a Y; c)- , . s,._, (u G]-,- . . . . .

If th= s~ce X is bicomlx~c~, and if d~m X < ~, [hen for n > dim X aH the ,spectra h n of this sequence of s p e c t r e a r e equal to ze ro . In th is c a s e condi t ion 3 of T h e o r e m 3 is s a t i s i f ed for the sequence of s p e c t r a . This ca~ be forn:uLated in the following w a y :

THEOREM 7. Let {X, Y) be a bicompact pair. Let dim X <-, I~*(X;G) be spectral hor.~ology groups; aml H*(X) be a spectral cohomology groups~ ~ sequences

.... ~1. (z ; G)-+ ~.(x; ~ -.. ~ . ( x . r ; G) -.- ~ . _ A y : ~ - , . . . . .

. . . . o, (# - * , (x ) , C)--, 0,( /~-* '{X. D " C) -+ 0 ' (~ ' (~0 , ~) -.- -.. 0, (~ - t " ) . 63 . . . .

a r e s e m i e x a c t tn the s e n s e that the compos i t i on o f two sequences of ho.momorphtsms L~ equal to ze ro .

Th is a l lows one to c o n s i d e r the homology g r o u p s of t he se sequences , that is t h e f ac to r g roups of k e r n e l s of homomorph i sms in the s equen c e s , in t e r m s of the images of the p r e c e d i n g h o m o m o r p h i s n ~ . The homology groups which a r e computed in t e r m s wr i t t en one below the o t h e r a r e ~ o m o r p h l c ,

An analogous p ropos i t ion is va l id for a s e q u e n c e of homology groups e r a f ini te d imens iona l b icompact ; t ra which corresL~Oncls to the exact s e q u e n c e 0 - ' G ~ - - _ . , - ~ ' - ' 0 of groups of coe f f i c i en t s ,

Another appl ica t ion o f the func tors P! i s r e l a t e d to the Borcl - .Moore homology g roup o f b ieompac t s p a c e s {1]. These homologies in the m e t r i c c a s e co inc ide wi'th the S teenrod-S i tn ikov g roups [5]. The B o r e l - M o o r e homologies Hn[X;G ) s a t i s f y the follo'~-ing u n i v e r s a l coeff ic ients formula=

o~ E~t ( # , - ( x ) . ~ ~ u . ( x ; c ) - , l~om ( ~ - (x ) . 63 --,-0. In view of the na tu ra l i t y o f th is fo rmula the fo l lowL~ commuta t ive d i a g r a m exis ts=

0--- Ext (li'" {X),G) --- 11. {X; G) -- llom (//" {X), G}--*O ~t el ti

0--- P Ext (h"". C)--~7t. (X; (;:)-.-. llom (Jr (X), C)

ZS2

The homomoL~,hl~m ~ h ~ Ltre~c,~ been ~tudicd in seetlon 9. It Is sa eplmorphtsm .~nd R~ kernel colx~.d~3 with a mayJ.k~al comp!cte s~up of the Ex~lin+1{X),~}: R follows from the dia~ram th~ the l~mo~rphlsm a is .~n e~orpblsm ~nd its ker~A coL~icide~ with a maxlrr~l complcte a~bgrm~

L I T E T e A T U R E C I T E D

L A. Borel, HomoloKy and DualRy in Genera~zed Manl[old~, Seminar on Tr~r~tormation g1"XmlS . Ann. Math. Stud. h~. _~, ;h-t~ceton (~60),

2. M.C.R. Butler a~-] G. Horrocks , C L ~ e s of Ex*~..nsions and Resolutions, ~11o~, Trans . Roy. Soc. London, A254. No. 1039, 155-222 (1.96i).

3. A. Grothendieck, Sur Qv.~e!que Points lYAlgebra Homologique, (tra.Rslzted i~to ]lusslan f rom) Tohuim Math. J . , H,, Moscow, 9, 119-221 (1957),

4. J . It. Roos, StL~ les Foncteurs I~ri@es de T.!mo Application, C . r . Acad. ScL, 25j._, No. 3702-3704 (1961).

5. N . E . Steenrod, l~e~a1,~r Cycles of Coml~act Metric Spaces, Ann. Math:, 41, No.4, 833--851 (L~

6. L. Fochs, AbelL~n Groups, Budapest (195S),

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