ON KUNNETH RELATIONS. Marek Golasiñski. the appropriate hypothesis the usual form of the Künneth re

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1 Pub. Mat. UAB Vol. 29 Ns Abrl 98 5 ON KUNNETH RELATIONS Marek Golasñsk Abstract. The am of ths ote s to subsume a umber of appretly qute dstrct results oe geeral theorem. For a left exact fuctor T : R-Mod --' Ab ad a cocha complex C * we gve a log exact sequece cludg the caocal map HT - TH, where H s the -th cohomology fuctor. Uder the approprate hypothess the usual form of the Küeth re lato (see [], chap. VI) s a specal case of our log exact sequece (Remark.2). Also the latest results of Coelho-Pezeec (see [2]) are cotaed ths log sequece (Proposto 2.3). I partcular, we obta a smple proof of the followg results of Osofsky o upper bouds of cohomologcal dmesos (see [7], [8)). If I s a drected set ad the cardal um ber of t s o greater tha 8m, the :..pdcolmir colmidi < supl.pdr M + m + the category of modules, 2. cd RcolmIG < supicd RG + m + the category of

2 groups.. Ma re sults. Let R be a rg ad let R-Mod ad Ab deote the category of left R-modules ad the category of abela groups, respectvely. For a left exact fuctor T : R-Mod Ab deote by T the rght -th derved fuctor of T ad for a cocha complex C * = (C,d ) of R-modules we put Z = Ker d, B = Im d - ad H = Z/B, whe- re rus over the tegers. Theorem.. (Geeral theorem). If TkC = 0 for k % ad all tegers, the there exsts a log exact sequece of abela groups 0 -> T Z - -+ HT -s TH - TH T Z > T3Z- -->.. --~ Tk Z --, TkB -~ Tk+2 Z - --~ Proof. The caocal short exact sequece of R-modules 0 -~ 7. -~ C -~ B+ - 0 yelds a log exact sequece of abela groups 0 -~ TZ Tl,~ TC T-d~ TB+ á0 T Z Tl TC Td --, TB+ j-.>... ak-~ TkZ Tk TkC Tkd TkB+l ak~,.. where TkB+l - 0 T k+l z s the coectg map.

3 By assumpto Tkc = 0 for k. : ad all tegers. So we get a short exact sequece + a a) 0 - TZ T> TC Td-~ TB -~4 TZ _ 0 ad a famly of somorphsms a b) TkB+ Ñk, Tk+Z for k > l ad all tegers The sequece a) yelds a somorphsm a') a TB+/m ~ T Z. Td Moreover, the caocal short exact sequece of R-modules 0 --> B -j, Z -É- ' H --> 0 yelds a log exact sequece of abela groups c) 0 - TB To yo TZ ---, TH --> TB TZ - Tl0 yl 7k- k Tk3 k TkB k 7k --> T H ->... --; T B ---~ T Z -~ T H -~..., where y k : T k H --> Tk+B s the usual coectg map. Hece, by b) we obta the followg log exact sequece of abela groups TR TZ a l 7 Tj(a-)- 0 -+/ TB ' TH T2 Z- T Z To a-ó y a -^~ Tk. (ak- ) - T H 2,...-k Tk+lZ-l - _ TkZ

4 The fuctor T : R-Mod -~ Ab s left exact, hece Ker Td = TZ ad the commutatve dagram 0 j H0 -' Im Td- ~ Ker Td T --> 0 Tj S ' 0 TB TZ TZ /TB 0 yelds Kero = cokerw = TB /I m Td-, by the Sake Lemma. : Kero --'> HT be the caocal cluso. The, fally, we obta the loa exact sequece 0 T Z- ~ - (a 0- ) Let -. (a -~ y HT T~ TH -0Y O` T2Z- " J o - 2 a - -~ TZ T ~` TH a 2 oyl> T3Z- T o 2 k. T(a-l ) - k a oy k+l j (a -- T 3 0 k TkZ T R ` Tk x k+l k, Tk+2 Z- -~ o. k+l As a corollary we get the usual form of the Küeth relato (see (, chap. VI). T R Remark ~'.2. If Tk = 0 for k > 2, the T Z TH ad the short sequece TH-, (a. - ) - o (T0 -l ) - TQ HT - TH _, 0

5 s exact. Moreover, the above Theorem yelds the followg result. Corollary.3. If the maps T k TkR k Z - T H duced by R the caocal map Z --> H are left splt (.e. there exsts a map p : TkÉ - 'TkZ such that p'ot'r' = dtkz ), the there exsts a log exact sequece of abela groups. _> T2k+lH kj... ~> T 3H-2, T H- -> HT -> TH.- T2H_ --> T4H-2 --~,. -> T2kH-k ->... Proof. I vrtue of assumpto the sequece c) from the proof of Theorem. determes the short exact sequece TR 7 0 l /TB - TH -> T B -> 0 ad the splt short exact sequeces Tk > TZ k F _ p k 7 k k+ T H f=~ T B -> 0 S for k > ad all tegers. Hece, usg the somorphsms b) from Theorem. we obta the followg dagram

6 -k-2 0-->T 2k+3 Z-k-28 -k ó (a 2k+2 )-T2k+ H -k- P.- k- T2k+Z-k- -, 0.-l a l 0-->T3 Z-2 8 ( 2 ) -> TH- -> T Z- -" 0 0-, TZ- t. a ( -l - ). HT - P > TZ /TB _, 0 - O~TZ ~- TH a- 0~ T2 Z- -, /TB 0 T ^ 0--. T2 Z- l , 2 - T H «3. 2. T 4 Z-2 ---, a , T4 Z-2 T!3 T4H , T6 Z-3, 0 -k -k T2kp-k a 2k+ y 2k -k * T2k H-k, T2k +2 Z-k-, 0 Composg the above short exact sequeces we obta the aouced log exact sequece of abela groups.o 2. Applcatos. Let I be a drected set. It sorell, kow that the fuctor colmi s exact. Moreover, f the cardal um ber of I s o greater tha t~m, the lme+k = 0 for k > 2 (see [4). Let. {R, ~pj }, j E I. ad {M P ` P j }, j E I be drected systems of rgs ad abela groups respectvely, such that

7 s each M a left R -module ad 0j (r M ) = w j (r ) Oj (M ) ad for r E R, m E M < j. (Such systems wll be called cou.vstet). The, M = colmim s a left R = colmim -module ad M :z; colm R M. the category of R-Mod. I R For further purposes the followg lemmas wll be useful. Lemma 2.. If each M s a (pure) projectve R -module for all E I, the lm HomR, (MN)^Ext(colmIMN)(^ Pext(colmIM,N)) for ay R-module N. Proof. A drected system {M,gL j },je I yelds a exact sequece of R-modules (see [3], Appedx I) --- < ~<, R M ~ R M. -~ R ~ M R0 0 0< R00 0I R00 ---' colmlr O. M^ colmim. s If each M a (pure) projectve R-module for all El the the above sequece s a R-(pure) projectve resoluto of colmim. Applyg the fuctor HomR (-,N) we obta the followg cha complexes :

8 0 0 HomR( É IR.M,N) R - HomR( l R MNO ) ~ (M É I HomR (M,N) ----->, l HomR.,N) --~ Cosequetly, lm HomR- (M,N) Ilt!Ext (colm IM,N) ( -Pe xt R (colm I M F N». Let FM, deotes the free R-module geerated by the elemets of M, the Fcolm M ~ colmifm * I Hece, we obta the followg geeralzato of Lemma 9.5 from [. Lemma 2.2. There exst R -(pure) projectve resolutos P of M formg a cosstet drected system {IP,.zj} such that IP= colmt IP~ s a R-(pure) projectve resoluto of colmim.,jei The two lemmas stated above wll be used the sequel. Let {C ;' ~j},j El be a cosstet drected system of cha complexes such that C are R.-modules for all E I. Put C * = colmlc* ad Zñ = coker dñ. The the followg geeralzato of the Coelho-Pezeec result s a smple cosequece of Theorem. ad Lemma 2.. Probosto 2.3. (see (2). If Cñ are (pure) projectve R -modules for all tegers, the the followg log 2

9 sequece (Z > lmi HomR.,N) - H (C,,N) - lml H (Cy,, N) --> lmí HomR,(Z,N) - lmí HomR. (ZñN) - lm, H'(C' N) lmí HomR. (Z,N) ->. lmé Ho (Z', N) - lmé H (C,,,N) -lm é+2 Hom (Z ',N)- I mr. I I R. - s exact. Moreover, as drect cosequeces of ths Proposto we obta the results of Osofsky (see [7] abd [8) ad Kelpfsk- Smso (see [ 6] ). Let.pdRM(.P.pdRP4) deote the left (pure) projectve dmeso of a R-module M ad let.9 dmr (.P.gl dmr) de ote the left (pure) global dmeso of a rg R. Corollary 2.4. ).pdcolm R, I colmim < supl.pdr. M + m + (.P.pdcolmIR colmim < supl.p.pdr M + m + ) ad ) l.gl dm colmim < supi.gl dmr 4 m +

10 (.P.gl dm colmir < supil.gl dmr + m + ). Proof. Applyg Proposto 2.3 to the drected sstem {Ip'~j }, j E I of projectve resolutos of {M, ~j}, j E I gve by Lemma 2.2 we obta the exact sequece 0 -- lml Hom (Z',N) --~ Ext (colm M. ---> lm Ext (M. N) - - I R R I M,N) R -+lm HomR (Zñ,N) -+ lm, HomR, (Zñ;N) --> lm, Ex tr. (M,,N) - --~ lmí HomR. (Zñ,N) - --> lmk Hom (Z'~',N) - lmé Ext.,N) - lmé 2 Hom (Z',N)- I R I R (M #N) -R where R = colmir. Hece, for > supi l.pdr. M morphsms we have the followg so- lmi HomR. (Zñ,N) - lmi HomR. (Zñ,N) lmi Hom R.(Zll,N) - lm,- Hom R.(Z,N). Therefore, for -k > sudi.pdr.m lmi HomR (Zr l l,n) lm, Hom R (Zñk,N).

11 But lmi = 0 for k > m +. Cosequetly, lmí HomR, (Zñ,N) = 0 ad Ex tr(colm IM, N) = 0 for > supl.pdr. M + m +. Hece,.pdcolmlR colml m < supl.pdrm + m +. ) For ay R-module M we have M = colmim, where M = M are R-modules for all ci. Therefore, by ). pdcolmirm _< supl. pdrm + m + < supil. g l dmr+ m + ad hece l.gl dm colmi R< supi.gl dmr + m +. The aalogous results for the left (pure) projectve ad global dmeso are obtaed by the same methods.n I partcular, f {G,spj },j El s a drected system of groups, the for group-rgs over a rg R we have R[ colmig - colmir[g l. So, by the above Corollary pdcolmir[gár < SuIpdR[ R + m +l, where OR deotes the trval module 66 over the approprate group-rg. Therefore, we get aother result due to Osofsky (see [8)

12 cd R colmig < sur) I cdrg + m +, where cd R deotes the R-cohomologcal dmeso. More geerally, f {C`pj},jE I s a drected system of small categores, thé usg methods smlar to those above, we obta cdr colmie < supi cdr C + m +. Remark 2.5. By results from 5 ad [9 we ca replace the drected set I by ay small category such that the fuctor colmi s exact. Now let R be a heredtary rg ad let{c ;,VGj ), j E I be a drected system of cha complexes over the category R-mod. Prooosto 2.6. If C* ad C ;* = colml C are cha complexes of projectve R-modules for all C=I, the the followg log sequece. lm k+l H-k- (C*, N) -,... --, lmi H-2 (C *,N) ---, lmi H- (C*,N) --' H (C*,N) - lmi H (C*,N) k -k -~ lmi H (C*,N) - lmi H (C *,N) ->... lmi H (C *,N)- s exact for all tegers ad ay R-module M.

13 Proof. Because {ZHomR(C*,N)}IEI = {HomR ( C/,N)}E I B ad the sequece j HC* C/ ---, /, -----> 0 splts, B C therefore {HomR(C/B,N) } E I - {HomR(HC*,,N) } El {HomR ( C/Z,N) } E I ad lmé ZHom (C N) = lmé Hom H (C N)) lmé Hom ( C,N). I R *' I R *' I R / Z C C But are projectve R-modules ad colml C/Z / Z = /Z s a projectve R-module, so by Lemma 2. lmí.homr (C/,N) = ExtR( C/ = 0 for k >. Z Z Moreover, by Uversal Coeffcet Theorem (see (]chap. VI) we have atural epmorphsms H (C*,N) ----» HomR(HC*,,Id) for all E I. Cosequetly, the map lmi Z HomR (C*, N) = lmihomr (H c*, N) -~ lmi H (C*,N) splts ad a approprate log exact sequece s determed by Corollary.2. Corollary 2.7. If {X, ~P j },j E I s a drected system of compact topologcal spaces, the the cocha fuctor commutes

14 wth lmts. Thus, the followg sequece of sgular cohomology groups 0 -> lm2 - H0 (x,a) --'. lma H-2 (X.A) --, lm H- (X,A) -----> H (X,A) -' lml H (X,A) -+ lm2 H- (X,A) - lml 4 H -2 I (X,A)--...-lm 2l H 0 (X,A) -', 0 s exact for ay abela group A, where X = colmix. Smlary, f {G,%Pj}, j E I ad {Cfspj},j El are drected systems of groups ad small categores respectvely, the we obta the approprate log exact sequece as above. REFERENCES [ Carta H., Eleberg S., "Homologcal algebra", Prceto, New Jersey (956). [2 Coelho M., Pezeec J., "Cohomologe d'ue lmte duct ve de complexes. Applcatos aux fatdmes de ftude e cohomologe tordue", C.R.Acad.Sc.Par s t.296. (983) p [3 Gabrel P., Zsma M., "Calculus fractos ad homotopy Theory", Sprger-Verlag, Berl (967). [4 - Goblot R., "Sur les derves de certaes lmtes project ves, applcatos aux modules", Bull. Sc.Math. (2) 94 (970) p [5 Isbell J., Mtchell B., "Exact colmts", Bull. Amer. Math.

15 [6 Kelpfsk R., Smso D., "O pure homolog.cal dmeso", Bull. Acad. Pol. Sc. XXIII, o.l, (975) p.l-6. [7] Osofsky B., "Upper bouds o, homologcal dmeso", Nagoya Math.J. 32(968) p [8], "Homologcal dmeso ad the cotuum hypothess", Tras.Amer.Math.Soc. 3 2 (968) p [9J Stauffer H.,B., "A relatoshp betwee left exact ad represetable fuctors", Ca.J.Math. XXIII, e.2 (97). Rebut el 30 de maíg del 984 Isttute of Mathematcs Ncholas Copercus Uversty Chopa 2/8, Toru, POLAND