Finitary Group Cohomology and Eilenberg-Mac Lane Spaces

FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES MAR TIN HAMIL TON Abstra t. W e sa y that a group G has  ohomolo gy almost ev- erywher e nitary if and only if the n th ohomology funtors of G omm ute with ltered olimits for all suien tly large n . In this pap er, w e sho w that if G is a group in Kropholler's lass LH F with ohomology almost ev erywhere nitary , then G has an Eilen b ergMa Lane spae K ( G, 1) whi h is dominated b y a CW- omplex with nitely man y n -ells for all suien tly large n . It is an op en question as to whether this holds for arbitrary G . W e also remark that the on v erse holds for an y group G . 1. Intr odution Let G b e a group and n ∈ N . The n th ohomology of G is a funtor H n ( G, − ) from the ategory of Z G -mo dules to the ategory of ab elian groups. W e are in terested in groups whose n th ohomology funtors are nitary ; that is, they omm ute with all ltered olimit systems of o eien t mo dules. W e are onerned with the lass LH F of lo ally hierar hially de- omp osable groups (see [10℄ for a denition of this lass). If G is an LH F -group, then Theorem 2.1 in [13℄ sho ws that the set { n ∈ N : H n ( G, − ) is nitary } is either onite or nite. If this set is onite, w e sa y that G has  ohomolo gy almost everywher e nitary , and if this set is nite, w e sa y that G has  ohomolo gy almost everywher e innitary . In [8℄ w e in v estigated algebrai  haraterisations of ertain lasses of LH F -groups with ohomology almost ev erywhere nitary . In this pap er w e pro v e the follo wing top ologial  haraterisation: Theorem A. L et G b e a gr oup in the lass LH F . Then the fol lowing ar e e quivalent: (i) G has  ohomolo gy almost everywher e nitary; 2000 Mathematis Subje t Classi ation. 20J06 55P20 18A22. Key wor ds and phr ases. ohomology of groups, nitary funtors, Eilen b ergMa Lane spaes. 1 2 MAR TIN HAMIL TON (ii) G × Z has an Eilenb er gMa L ane sp a e K ( G × Z , 1) with nitely many n - el ls for al l suiently lar ge n ; (iii) G has an Eilenb er gMa L ane sp a e K ( G, 1) whih is domi- nate d by a CW- omplex with nitely many n - el ls for al l su- iently lar ge n . The impliations (ii) ⇒ (iii) and (iii) ⇒ (i) hold for an y group G , while our pro of of (i) ⇒ (ii) requires the assumption that G b elongs to the lass LH F . W e do not kno w whether (i) ⇒ (ii) holds for arbitrary G . 1.1. A  kno wledgemen ts. I w ould lik e to thank m y resear h sup er- visor P eter Kropholler for suggesting that a result lik e Theorem A should b e true, and for his advie and supp ort throughout this pro jet. I w ould also lik e to thank Philipp Reinhard for explaining the argu- men ts in Lemma 2.22 2. Pr oof 2.1. Pro of of Theorem A (i) ⇒ (ii). Supp ose that G is an LH F -group with ohomology almost ev erywhere nitary . W e need to mak e use of  omplete  ohomolo gy , and refer the reader to [5, 6, 16℄ for further information on denitions et. If R is a ring, then w e an onsider the stable  ate gory of R -mo dules; the ob jets are the R -mo dules and the stable maps M → N b et w een R -mo dules are the elemen ts of the omplete ohomology group d Ext 0 R ( M , N ) . W e mak e the follo wing denitions: Denition 2.1. Let R b e a ring. An R -mo dule M is said to b e  om- pletely nitary (o v er R ) if and only if the funtor d Ext n R ( M , − ) is nitary for all in tegers n . R emark 2.2 . W e see from 4 . 1 (ii) in [10℄ that ev ery R -mo dule of t yp e FP ∞ is ompletely nitary . Denition 2.3. Let R b e a ring. An R -mo dule N is said to b e  om- pletely at (o v er R ) if and only if d Ext 0 R ( M , N ) = 0 for all ompletely nitary R -mo dules M . W e ha v e a v ersion of the E kmannShapiro Lemma for omplete ohomology (Lemma 1 . 3 in [12℄): FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 3 Lemma 2.4. L et H b e a sub gr oup of G , V b e a Z H -mo dule and N b e a Z G -mo dule. Then, for al l inte gers n , ther e is a natur al isomorphism d Ext n Z G ( V ⊗ Z H Z G, N ) ∼ = d Ext n Z H ( V , N ) . No w, let G b e an LH F -group, and N b e a Z G -mo dule. T o  he k whether N is ompletely at, it is enough to  he k whether the re- strition of N to ev ery nite subgroup of G is ompletely at, b y the follo wing prop osition. This is where the assumption that G b elongs to LH F is used. Prop osition 2.5. L et G b e an LH F -gr oup, and N b e a Z G -mo dule. Then the fol lowing ar e e quivalent: (i) N is  ompletely at as a Z G -mo dule; (ii) N is  ompletely at as a Z K -mo dule for al l nite sub gr oups K of G . Pr o of. • (i) ⇒ (ii): F ollo ws from Lemma 2.4 . • (ii) ⇒ (i): An easy generalization of Prop osition 6 . 8 in [12 ℄ sho ws that if N is a Z G -mo dule whi h is ompletely at as a Z K -mo dule for all nite subgroups K of G , then N ⊗ Z H Z G is ompletely at as a Z G -mo dule for all LH F -subgroups H of G . Then, as w e are assuming that G b elongs to LH F , the result follo ws.  W rite B := B ( G, Z ) for the Z G -mo dule of b ounded funtions from G to Z . F ollo wing Benson [1 , 2℄, a Z G -mo dule M is said to b e  obr ant if M ⊗ B is pro jetiv e. W e no w mak e the follo wing denition: Denition 2.6. Let G b e a group. A Z G -mo dule is alled b asi if it is of the form U ⊗ Z K Z G , where K is a nite subgroup of G and U is a ompletely nitary , obran t Z K -mo dule. A Z G -mo dule M is alled p oly-b asi if it has a series 0 = M 0 ≤ · · · ≤ M n = M in whi h the setions M i / M i − 1 are basi. The rst step in the pro of of Theorem A in v olv es the follo wing on- strution, whi h is a v ariation on the onstrution found in  4 of [12 ℄: Denition 2.7. Let G b e a group, and M b e a Z G -mo dule. W e onstrut a  hain M = M 0 ⊆ M 1 ⊆ M 2 ⊆ · · · 4 MAR TIN HAMIL TON indutiv ely so that for ea h n ≥ 0 there is a short exat sequene C n ֌ M n ⊕ P n ։ M n +1 in whi h (i) C n is a diret sum of basi mo dules; (ii) P n is pro jetiv e; and (iii) ev ery map from a basi mo dule to M n fators through C n . Set M 0 = M . Supp ose that n ≥ 0 and that M n has b een onstruted. Consider the p oin ted ategory whose ob jets are ordered pairs ( C , φ ) , where C is a basi mo dule and φ is a homomorphism from C to M n , and whose morphisms are the ob vious omm utativ e triangles. Cho ose a set X n on taining at least one ob jet of this ategory from ea h iso- morphism lass. Set C n := L ( C,φ ) ∈ X n C and use the maps φ asso iated to ea h ob jet to dene a map C n → M n . Prop erties (i) and (iii) are no w guaran teed. Note that an y basi mo dule U ⊗ Z K Z G an b e written as a diret sum of opies of U . Then, as tensor pro duts omm ute with diret sums, w e see that an y basi mo dule is itself obran t. Hene C n is obran t, so C n ⊗ B is pro jetiv e and w e an set P n := C n ⊗ B . Finally , M n +1 an b e dened as the ok ernel of this inlusion C n → M n ⊕ P n , or in other w ords the pushout, and sine the map C n → P n is an inlusion, it follo ws that the indued map M n → M n +1 is also injetiv e and w e regard M n as a submo dule of M n +1 . Finally , set M ∞ to b e the olimit M ∞ := lim − → n M n . Next, w e ha v e the follo wing te hnial prop osition, whi h shall b e needed in the pro of of Prop osition 2.13 : Prop osition 2.8. L et G b e a gr oup, and M b e a Z G -mo dule. Con- strut the hain M = M 0 ⊆ M 1 ⊆ M 2 ⊆ · · · as in Denition 2.7 . Then for e ah n , we  an expr ess M n +1 as a lter e d  olimit M n +1 := lim − → λ n M n ⊕ P λ n C λ n wher e P λ n is pr oje tive and C λ n is p oly-b asi. Pr o of. Let X n b e the set dened in Denition 2.7. W e an write X n as the ltered olimit of its nite subsets X n := lim − → λ n X λ n . FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 5 Set C λ n := M ( C,φ ) ∈ X λ n C , and P λ n := C λ n ⊗ B . The result no w follo ws.  The next step in the pro of is to sho w that the mo dule M ∞ is om- pletely at. Reall (see, for example,  3 of [1℄) that if M and N are Z G - mo dules, then Hom Z G ( M , N ) is the quotien t of Hom Z G ( M , N ) b y the additiv e subgroup onsisting of homomorphisms whi h fator through a pro jetiv e mo dule. W e ha v e the follo wing useful result (Lemma 2 . 3 in [12℄): Lemma 2.9. L et M and N b e Z G -mo dules. If M is  obr ant, then the natur al map Hom Z G ( M , N ) → d Ext 0 Z G ( M , N ) is an isomorphism. W e also need the follo wing lemma: Lemma 2.10. L et G b e a nite gr oup, and V b e a Z G -mo dule. Then V is  obr ant if and only if V is fr e e as a Z -mo dule. Pr o of. Let B := B ( G, Z ) denote the Z G -mo dule of b ounded funtions from G to Z . First, note that as G is a nite group, B ∼ = Z G . Supp ose that V is free as a Z -mo dule. Then V ⊗ B ∼ = V ⊗ Z G is free as a Z G -mo dule, and hene V is obran t. Con v ersely , supp ose that V is obran t, so V ⊗ B ∼ = V ⊗ Z G is a pro jetiv e Z G -mo dule. Then V ⊗ Z G is pro jetiv e as a Z -mo dule, but as Z is a prinipal ideal domain, ev ery pro jetiv e Z -mo dule is free. Hene, V ⊗ Z G is free as a Z -mo dule, and so it follo ws that V is free as a Z -mo dule.  W e an no w pro v e that M ∞ is ompletely at: Lemma 2.11. L et G b e an LH F -gr oup, and M b e any Z G -mo dule. Then the mo dule M ∞ ,  onstrute d as in Denition 2.7, is  ompletely at. Pr o of. This is a generalization of Lemma 4 . 1 in [12℄: As G b elongs to LH F , w e see from Prop osition 2.5 that it is enough to sho w that M ∞ is ompletely at o v er Z K for all nite subgroups K of G . By Lemma 2.4 it is then enough to sho w that d Ext 0 Z G ( U ⊗ Z K Z G, M ∞ ) = 0 6 MAR TIN HAMIL TON for ev ery nite subgroup K of G and ev ery ompletely nitary Z K - mo dule U . Fix K and U . As K is nite, U has a omplete resolution in the sense of [6 ℄. Let V b e the zeroth k ernel in one su h resolution, so V is a submo dule of a pro jetiv e Z K -mo dule. Therefore, V is free as a Z -mo dule and it then follo ws from Lemma 2.10 that V is obran t as a Z K -mo dule. Then, as U is stably isomorphi to V , (that is, U and V are isomorphi as ob jets of the stable ategory of Z K -mo dules, dened at the b eginning of this setion), it is enough to pro v e that d Ext 0 Z G ( V ⊗ Z K Z G, M ∞ ) = 0 . Therefore, w e only need to sho w that d Ext 0 Z G ( C , M ∞ ) = 0 for all basi Z G -mo dules C . Let C b e a basi Z G -mo dule. As C is obran t, it follo ws from Lemma 2.9 that the natural map Hom Z G ( C , M ∞ ) → d Ext 0 Z G ( C , M ∞ ) is an isomorphism. Let φ ∈ Hom Z G ( C , M ∞ ) . As C is basi, it is ompletely nitary , and w e see that the natural map lim − → n Hom Z G ( C , M n ) → Hom Z G ( C , M ∞ ) is an isomorphism. Therefore, w e an view φ as an elemen t of lim − → n Hom Z G ( C , M n ) , and so φ is represen ted b y some e φ ∈ Ho m Z G ( C , M n ) for some n . Then, as the follo wing diagram omm utes: Hom Z G ( C , M n ) / / ) ) S S S S S S S S S S S S S S S lim − → n Hom Z G ( C , M n )   Hom Z G ( C , M ∞ ) w e see that φ is in fat the image of e φ under the map Hom Z G ( C , ι ) : Hom Z G ( C , M n ) → Hom Z G ( C , M ∞ ) indued b y the natural map ι : M n → M ∞ . The image Hom Z G ( C , ι )( e φ ) is dened as follo ws: As e φ ∈ Ho m Z G ( C , M n ) , it is represen ted b y some map α : C → M n . W e an then onsider the map f : C α → M n ι → M ∞ . Let f denote the image of f in Hom Z G ( C , M ∞ ) . Then Hom Z G ( C , ι )( e φ ) := f . FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 7 No w, b y onstrution, w e see that the omp osite C → M n ֒ → M n +1 fators through the pro jetiv e mo dule P n . Hene, f fators through a pro jetiv e, and so f = 0 . W e then onlude that Hom Z G ( C , M ∞ ) = 0 , and so d Ext 0 Z G ( C , M ∞ ) = 0 , and therefore M ∞ is ompletely at o v er Z G , as required.  Next, reall the follo wing v ariation on S han uel's Lemma (Lemma 3 . 1 in [12 ℄): Lemma 2.12. L et M ′′ ι ֌ M π ։ M ′ b e any short exat se quen e of R -mo dules in whih π fators thr ough a pr oje tive mo dule Q . Then M is isomorphi to a dir e t summand of Q ⊕ M ′′ . W e no w use the fat that the Z G -mo dule M ∞ is ompletely at to pro v e the follo wing: Prop osition 2.13. L et G b e an LH F -gr oup and M b e a  ompletely ni- tary,  obr ant Z G -mo dule. Then M is isomorphi to a dir e t summand of the dir e t sum of a p oly-b asi mo dule and a pr oje tive mo dule. Pr o of. This is a generalization of an argumen t found in  4 of [12 ℄: As in Denition 2.7 , onstrut the  hain M = M 0 ⊆ M 1 ⊆ M 2 ⊆ · · · of Z G -mo dules, and let M ∞ := lim − → n M n . As G b elongs to LH F , w e see from Lemma 2.11 that M ∞ is ompletely at, and so d Ext 0 Z G ( M , M ∞ ) = 0 . Also, as M is obran t, it follo ws from Lemma 2.9 that Hom Z G ( M , M ∞ ) = 0 . Then, as M is ompletely nitary , w e see that lim − → n Hom Z G ( M , M n ) = 0 . Therefore, there m ust b e some n su h that the iden tit y map on M maps to zero in Hom Z G ( M , M n ) . Hene, w e see that the inlusion M ֒ → M n fators through a pro jetiv e mo dule. By Prop osition 2.8, w e an write M n as a ltered olimit, M n = lim − → λ n − 1 M n − 1 ⊕ P λ n − 1 C λ n − 1 := lim − → λ n − 1 M λ n − 1 , 8 MAR TIN HAMIL TON where ea h P λ n − 1 is pro jetiv e and ea h C λ n − 1 is p oly-basi. Then, as M is ompletely nitary , a similar argumen t to ab o v e sho ws that there is some λ n − 1 su h that the inlusion M ֒ → M λ n − 1 fators through a pro jetiv e mo dule. No w, w e an also write M λ n − 1 as a ltered olimit: M λ n − 1 = lim − → λ n − 2 ( M n − 2 ⊕ P λ n − 2 C λ n − 2 ) ⊕ P λ n − 1 C λ n − 1 := lim − → λ n − 2 M λ n − 2 , and w e on tin ue as ab o v e. Con tin uing in this w a y , w e ev en tually obtain a map M ֒ → M λ 0 whi h fators through a pro jetiv e mo dule Q . No w, M λ 0 has b een onstruted in su h a w a y that w e ha v e a short exat sequene K ֌ M ⊕ P ։ M λ 0 , where P := P λ 0 ⊕ · · · ⊕ P λ n − 1 , and K admits a ltration 0 = K − 1 ≤ K 0 ≤ · · · ≤ K n − 1 = K , with ea h K i /K i − 1 isomorphi to C λ i . W e see that the seond map in the ab o v e short exat sequene m ust fator through P ⊕ Q , and as K is learly p oly-basi, the result no w follo ws from Lemma 2.12 .  W e an no w pro v e the follo wing: Prop osition 2.14. L et G b e an LH F -gr oup, and M b e a  ompletely nitary,  obr ant Z G -mo dule. Then M is isomorphi to a dir e t sum- mand of a Z G -mo dule whih has a pr oje tive r esolution that is eventu- al ly nitely gener ate d. Pr o of. W e b egin b y sho wing that basi Z G -mo dules are isomorphi to diret summands of Z G -mo dules with pro jetiv e resolutions that are ev en tually nitely generated. Reall that basi Z G -mo dules are of the form U ⊗ Z K Z G , where K is a nite subgroup of G and U is a ompletely nitary , obran t Z K -mo dule. W rite U as the ltered olimit of its nitely presen ted submo dules, U = lim − → λ U λ . As U is ompletely nitary and obran t, it follo ws that Hom Z K ( U, − ) is nitary , and so the natural map lim − → λ Hom Z K ( U, U /U λ ) → Hom Z K ( U, lim − → λ U /U λ ) is an isomorphism; that is, lim − → λ Hom Z K ( U, U /U λ ) = 0 . FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 9 Therefore, there m ust b e some λ su h that the iden tit y map on U maps to zero in Hom Z K ( U, U /U λ ) . Hene, w e see that the surjetion U ։ U /U λ fators through a pro jetiv e Z K -mo dule Q . Then, b y Lemma 2.12 , w e see that U is isomorphi to a diret summand of Q ⊕ U λ . No w, as K is nite, ev ery nitely presen ted Z K -mo dule is of t yp e FP ∞ , so in partiular U λ is of t yp e FP ∞ . Then, as U ⊗ Z K Z G is isomorphi to a diret summand of Q ⊗ Z K Z G ⊕ U λ ⊗ Z K Z G , where Q ⊗ Z K Z G is pro jetiv e, and U λ ⊗ Z K Z G is of t yp e FP ∞ , w e see that U ⊗ Z K Z G is isomorphi to a diret summand of a Z G -mo dule with a pro jetiv e resolution that is ev en tually nitely generated. Next, as p oly-basi mo dules are built up from basi mo dules b y extensions, w e see from the Horsesho e Lemma that ev ery p oly-basi Z G -mo dule is isomorphi to a diret summand of a Z G -mo dule with a pro jetiv e resolution that is ev en tually nitely generated. Finally , if G is an LH F -group, and M is a ompletely nitary , o- bran t Z G -mo dule, it follo ws from Prop osition 2.13 that M is isomor- phi to a diret summand of P ⊕ C , for some pro jetiv e mo dule P and some p oly-basi mo dule C . Then, as C is isomorphi to a diret sum- mand of a Z G -mo dule with a pro jetiv e resolution that is ev en tually nitely generated, the result no w follo ws.  W e no w ha v e the follo wing prop osition: Prop osition 2.15. L et G b e an LH F -gr oup, and M b e a  ompletely nitary Z G -mo dule. A lso, let B := B ( G, Z ) denote the Z G -mo dule of b ounde d funtions fr om G to Z . Then M ⊗ B has nite pr oje tive dimension over Z G . Pr o of. This is a generalization of Prop osition 9 . 2 in [ 5 ℄: Let K b e a nite subgroup of G . W e see from the Prop osition in [15 ℄ that B is free as a Z K -mo dule, so M ⊗ B is a diret sum of opies of M ⊗ Z K as a Z K -mo dule, and hene has nite pro jetiv e dimension o v er Z K . It then follo ws from Lemma 4 . 2 . 3 in [11℄ that d Ext 0 Z K ( A, M ⊗ B ) = 0 for an y Z K -mo dule A . In partiular, w e see that M ⊗ B is ompletely at o v er Z K . As this holds for an y nite subgroup K of G , w e see from Prop osition 2.5 that M ⊗ B is ompletely at o v er Z G . Then, as M is ompletely nitary o v er Z G , w e see that d Ext 0 Z G ( M , M ⊗ B ) = 0 , and it then follo ws from Lemma 2 . 2 in [7℄ that M ⊗ B has nite pro- jetiv e dimension o v er Z G .  10 MAR TIN HAMIL TON Lemma 2.16. L et G b e an LH F -gr oup with  ohomolo gy almost every- wher e nitary. Then ther e is an inte ger n ≥ 0 suh that in any pr o- je tive r esolution P ∗ ։ Z of G the n th kernel is a  ompletely nitary,  obr ant mo dule. Pr o of. As G has ohomology almost ev erywhere nitary , it follo ws from 4 . 1 (ii) in [10℄ that the trivial Z G -mo dule Z and ev ery k ernel of a pro je- tiv e resolution of G is ompletely nitary . By Prop osition 2.15 it follo ws that B has nite pro jetiv e dimension o v er Z G . If pro j . dim Z G B = n , then learly the n th k ernel of an y pro jetiv e resolution of G is o- bran t.  Next, w e ha v e t w o straigh tforw ard results: Prop osition 2.17. L et R b e a ring, and supp ose that 0 → N ′ → N → P n → · · · → P 0 → M → 0 is an exat se quen e of R -mo dules suh that the P i ar e pr oje tive, and N ′ and N have pr oje tive r esolutions that ar e eventual ly nitely gener- ate d. Then the p artial pr oje tive r esolution P n → · · · → P 0 → M → 0 of M  an b e extende d to a pr oje tive r esolution that is eventual ly nitely gener ate d. Pr o of. Let K := Ker( P n → P n − 1 ) , so w e ha v e the follo wing short exat sequene: N ′ ֌ N ։ K . Next, let Q ∗ ։ N b e a pro jetiv e resolution of N that is ev en tually nitely generated, and let L denote the zeroth k ernel. W e then ha v e the follo wing: e K A A A A L     Q 0     B B B B N ′ / / / / N / / / / K where e K is an extension of N ′ b y L , and sine b oth N ′ and L ha v e pro jetiv e resolutions that are ev en tually nitely generated, it follo ws from the Horsesho e Lemma that e K also has su h a resolution. W e then ha v e the follo wing exat sequene: 0 → e K → Q 0 → P n → · · · → P 0 → M → 0 , FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 11 and the result no w follo ws.  Prop osition 2.18. L et M b e an R -mo dule. If M has a pr oje tive r eso- lution that is eventual ly nitely gener ate d, then M has a fr e e r esolution that is eventual ly nitely gener ate d. Pr o of. Let P ∗ ։ M b e a pro jetiv e resolution of M that is ev en tually nitely generated; sa y P j is nitely generated for all j ≥ n , and let K := Ker( P n − 1 → P n − 2 ) . Then K is of t yp e FP ∞ , and hene of t yp e FL ∞ . W e an therefore  ho ose a free resolution F n + ∗ ։ K of K with all the free mo dules nitely generated. This giv es the follo wing exat sequene: · · · → F n +1 → F n → P n − 1 → · · · → P 2 → P 1 → P 0 → M → 0 . Next, reall the Eilen b erg tri k (Lemma 2 . 7 VI I I in [4℄): F or an y pro jetiv e R -mo dule P , w e an  ho ose a free R -mo dule F su h that P ⊕ F ∼ = F . Therefore, using this, w e an replae the pro jetiv e mo dules P i in the ab o v e exat sequene b y free mo dules F i , at the exp ense of  hanging F n to a larger free mo dule F ′ n . W e then ha v e the follo wing free resolution · · · → F n +2 → F n +1 → F ′ n → F n − 1 → · · · → F 0 → M → 0 of M , with the F j nitely generated for all j ≥ n + 1 .  W e no w ha v e the follo wing prop osition (Prop osition 5 . 1 in [14℄): Prop osition 2.19. L et X n b e an ( n − 1) - onne te d n -dimensional G - CW- omplex, wher e n ≥ 2 . L et φ : F → H n ( X n ) b e a surje tive Z G -mo dule map fr om a fr e e Z G -mo dule F to the n th homolo gy of X n . Then X n  an b e emb e dde d into an n - onne te d ( n + 1) -dimensional G - CW- omplex X n +1 suh that G ats fr e ely outside X n and ther e is a short exat se quen e 0 → H n +1 ( X n +1 ) → F → H n ( X n ) → 0 . Finally , w e an no w pro v e the impliation (i) ⇒ (ii) of Theorem A. Theorem 2.20. L et G b e an LH F -gr oup with  ohomolo gy almost ev- erywher e nitary. Then G × Z has an Eilenb er gMa L ane sp a e K ( G × Z , 1) with nitely many n - el ls for al l suiently lar ge n . Pr o of. Let Y b e the 2 -omplex asso iated to some presen tation of G , and let e Y denote its univ ersal o v er. The augmen ted ellular  hain omplex of e Y is a partial free resolution of the trivial Z G -mo dule, whi h w e denote b y F 2 → F 1 → F 0 → Z → 0 . 12 MAR TIN HAMIL TON W e an extend this to a free resolution F ∗ ։ Z of the trivial Z G - mo dule, and as G is an LH F -group with ohomology almost ev erywhere nitary , it follo ws from Lemma 2.16 that there is some m ≥ 0 su h that the m th k ernel M := Ker( F m − 1 → F m − 2 ) of this resolution is a ompletely nitary , obran t Z G -mo dule. W e then ha v e the follo wing exat sequene of Z G -mo dules: 0 → M → F m − 1 → · · · → F 0 → Z → 0 . Next, reall that the irle S 1 is an Eilen b ergMa Lane spae K ( Z , 1) , with univ ersal o v er R . The augmen ted ellular  hain omplex of R is the follo wing free resolution of the trivial ZZ -mo dule: 0 → ZZ → ZZ → Z → 0 . If w e tensor these t w o exat sequenes together, w e obtain the fol- lo wing exat sequene of Z [ G × Z ] -mo dules: 0 → M ⊗ ZZ → M ⊗ ZZ ⊕ F m − 1 ⊗ ZZ → F m − 1 ⊗ ZZ ⊕ F m − 2 ⊗ ZZ → · · · → F 0 ⊗ ZZ → Z → 0 . No w, as M is a ompletely nitary , obran t Z G -mo dule, it follo ws from Prop osition 2.14 that M is isomorphi to a diret summand of some Z G -mo dule L whi h has a pro jetiv e resolution that is ev en tually nitely generated. W e then obtain the follo wing exat sequene of Z [ G × Z ] -mo dules: 0 → L ⊗ ZZ → L ⊗ ZZ ⊕ F m − 1 ⊗ ZZ → F m − 1 ⊗ ZZ ⊕ F m − 2 ⊗ ZZ → · · · → F 0 ⊗ ZZ → Z → 0 . It no w follo ws from Prop ositions 2.17 and 2.18 that w e an extend the partial free resolution F m − 1 ⊗ ZZ ⊕ F m − 2 ⊗ ZZ → · · · → F 0 ⊗ ZZ → Z → 0 of the trivial Z [ G × Z ] -mo dule to a free resolution that is ev en tually nitely generated. W e shall denote this free resolution b y F ′ ∗ ։ Z . Next, let X 2 denote the sub omplex of e Y × R , onsisting of the 0 , 1 and 2 -ells. Then, as C ∗ ( e Y × R ) ∼ = C ∗ ( e Y ) ⊗ C ∗ ( R ) , w e see that the augmen ted ellular  hain omplex of X 2 is the follo wing: F ′ 2 → F ′ 1 → F ′ 0 → Z → 0 , FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 13 and, furthermore, that e H i ( X 2 ) = 0 for i = 0 , 1 . W e therefore ha v e the follo wing exat sequene: 0 → e H 2 ( X 2 ) → F ′ 2 → F ′ 1 → F ′ 0 → Z → 0 , and as F ′ 3 ։ e H 2 ( X 2 ) , it follo ws from Prop osition 2.19 that w e an em b ed X 2 in to a 2 -onneted 3 -omplex X 3 su h that w e ha v e the follo wing short exat sequene: 0 → e H 3 ( X 3 ) → F ′ 3 → e H 2 ( X 2 ) → 0 . Then F ′ 4 ։ e H 3 ( X 3 ) , and w e an on tin ue as b efore. By indution, w e an then onstrut a spae, whi h w e denote b y X , su h that C n ( X ) = F ′ n for all n . Then, as the free resolution F ′ ∗ ։ Z is ev en tually nitely generated, it follo ws that C n ( X ) is nitely generated for all suien tly large n . Also, w e see that e H i ( X ) = 0 for all i , and so X is on tratible (see I. 4 in [4℄). W e see from Prop osition 1 . 40 in [9℄ that X is the univ ersal o v er for the quotien t spae X := X/ G × Z , and furthermore that X has fundamen tal group isomorphi to G × Z . Th us, X is an Eilen b erg Ma Lane spae K ( G × Z , 1) , and as C n ( X ) is nitely generated for all suien tly large n , w e onlude that X has nitely man y n -ells for all suien tly large n , as required.  2.2. Pro of of Theorem A (ii) ⇒ (iii). W e do not require the assumption that G b elongs to LH F for this setion. Reall from page 528 of [9℄ that a spae Y is said to b e dominate d b y a spae K if and only if Y is a retrat of K in the homotop y ategory; that is, there are maps i : Y → K and r : K → Y su h that r i ≃ id Y . Prop osition 2.21. Supp ose that K is a K ( G × Z , 1) sp a e with nitely many n - el ls for al l suiently lar ge n . Then G has an Eilenb er gMa L ane sp a e K ( G, 1) whih is dominate d by K . Pr o of. As ev ery group has an Eilen b ergMa Lane spae (Theorem 7 . 1 VI I I in [4℄), w e an  ho ose a K ( G, 1) spae Y . Then, as S 1 is a K ( Z , 1) spae, w e see from Example 1 B. 5 in [9℄ that Y × S 1 is a K ( G × Z , 1) spae. Then, as K ( G × Z , 1) spaes are unique up to homotop y equiv alene (Theorem 1 B. 8 in [9℄), w e see that Y × S 1 ≃ K , and hene that Y is dominated b y K .  2.3. Pro of of Theorem A (iii) ⇒ (i). One again, w e do not require the assumption that G b elongs to LH F for this setion. 14 MAR TIN HAMIL TON Lemma 2.22. L et Y b e a K ( G, 1) sp a e whih is dominate d by a CW- omplex with nitely many  el ls in al l suiently high dimensions. Then we may ho ose this  omplex to have fundamental gr oup isomor- phi to G . Pr o of. Let Y b e dominated b y a CW-omplex K that has nitely man y ells in all suien tly high dimensions, so there are maps Y i → K r → Y su h that r i ≃ id Y . Applying π 1 giv es maps π 1 ( Y ) π 1 ( i ) → π 1 ( K ) π 1 ( r ) → π 1 ( Y ) su h that π 1 ( r ) π 1 ( i ) = id π 1 ( Y ) . Hene, π 1 ( r ) is surjetiv e. Let K ′ denote the k ernel of π 1 ( r ) , and let W b e a b ouquet of irles, with one irle for ea h generator in some  hosen presen tation of K ′ , so there is an ob vious map W → K . Next, let C W denote the one on W , and form the follo wing pushout: W / /   K      C W / / _ _ _ L It follo ws that L is a CW-omplex with nitely man y ells in all su- ien tly high dimensions. No w, the omp osite map W → K r → Y is learly n ullhomotopi, and therefore lifts through the one, so w e ha v e the follo wing diagram: W / /   K   r   C W / / , , L Y and so b y the denition of pushout, there is an indued map L → Y making the ab o v e diagram omm ute. If w e no w omp ose this with the map Y i → K → L , w e obtain a map Y → L → Y that is homotopi to the iden tit y on Y . Hene, Y is dominated b y L . Finally , b y v an Kamp en's Theorem (Theorem 1 . 20 in [ 9℄), w e see that π 1 ( L ) ∼ = π 1 ( K ) / Im( π 1 ( W ) → π 1 ( K )) ∼ = π 1 ( Y ) ∼ = G, FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 15 as required.  Next, reall from [ 3℄ that if P := ( P i ) i ≥ 0 is a  hain omplex of pro- jetiv e Z G -mo dules, then w e dene the  ohomolo gy the ory H ∗ ( P , − ) determine d by P as H n ( P , M ) := H n (Hom Z G ( P ∗ , M )) for ev ery Z G -mo dule M and ev ery n ∈ N . Lemma 2.23. L et P := ( P i ) i ≥ 0 b e a hain  omplex of pr oje tive Z G - mo dules. If P n − 1 , P n and P n +1 ar e nitely gener ate d, then H n ( P , − ) is nitary. Pr o of. Firstly , reall from Lemma 4 . 7 VI I I in [4℄ that if Q is a nitely generated pro jetiv e mo dule, then the funtor Hom Z G ( Q, − ) is nitary . Next, let M := Cok er ( P n +1 → P n ) , so w e ha v e the follo wing exat sequene: P n +1 → P n → M → 0 , whi h giv es the follo wing exat sequene of funtors: 0 → Hom Z G ( M , − ) → Hom Z G ( P n , − ) → Hom Z G ( P n +1 , − ) . Let ( N λ ) b e an y ltered olimit system of Z G -mo dules, so w e ha v e the follo wing omm utativ e diagram with exat ro ws: 0 / /   lim − → λ Hom Z G ( M , N λ ) / /   lim − → λ Hom Z G ( P n , N λ ) / /   lim − → λ Hom Z G ( P n +1 , N λ )   0 / / Hom Z G ( M , lim − → λ N λ ) / / Hom Z G ( P n , lim − → λ N λ ) / / Hom Z G ( P n +1 , lim − → λ N λ ) and as b oth Hom Z G ( P n , − ) and Hom Z G ( P n +1 , − ) are nitary , the t w o righ t-hand maps are isomorphisms. It then follo ws from the Fiv e Lemma that the natural map lim − → λ Hom Z G ( M , N λ ) → Hom Z G ( M , lim − → λ N λ ) is an isomorphism, and hene that Hom Z G ( M , − ) is nitary . Then, as w e ha v e the follo wing exat sequene of funtors: Hom Z G ( P n − 1 , − ) → Hom Z G ( M , − ) → H n ( P , − ) → 0 , the result no w follo ws from another appliation of the Fiv e Lemma.  W e an no w pro v e the impliation (iii) ⇒ (i) of Theorem A: 16 MAR TIN HAMIL TON Prop osition 2.24. Supp ose that G has an Eilenb er gMa L ane sp a e K ( G, 1) whih is dominate d by a CW- omplex with nitely many n - el ls for al l suiently lar ge n . Then G has  ohomolo gy almost everywher e nitary. Pr o of. This is a generalization of the pro of of Prop osition 6 . 4 VI I I in [4 ℄: Let Y b e su h a K ( G, 1) spae. By Lemma 2.22 , w e see that Y is dominated b y a CW-omplex K with nitely man y ells in all su- ien tly high dimensions, su h that K has fundamen tal group isomor- phi to G . Let e Y and e K denote the resp etiv e univ ersal o v ers. W e see that C ∗ ( e Y ) is a retrat of C ∗ ( e K ) in the homotop y ategory of  hain omplexes o v er Z G . Therefore, w e obtain maps giving the follo wing omm utativ e diagram: H ∗ ( G, − ) N N N N N N N N N N N N N N N N N N N N N N / / H ∗ ( C , − )   H ∗ ( G, − ) where H ∗ ( C , − ) denotes the ohomology theory determined b y C ∗ ( e K ) . W e then onlude that H ∗ ( G, − ) is a diret summand of H ∗ ( C , − ) . No w, as K has nitely man y ells in all suien tly high dimensions, it follo ws that C ∗ ( e K ) is ev en tually nitely generated, and so b y Lemma 2.23 that H k ( C , − ) is nitary for all suien tly large k . The result then follo ws from an appliation of the Fiv e Lemma.  Referenes [1℄ D. J. Benson, Complexity and varieties for innite gr oups. I , J. Algebra 193 (1997), no. 1, 260287. [2℄ , Complexity and varieties for innite gr oups. II , J. Algebra 193 (1997), no. 1, 288317. [3℄ Kenneth S. Bro wn, Homolo gi al riteria for niteness , Commen t. Math. Helv. 50 (1975), 129135. [4℄ , Cohomolo gy of gr oups , Graduate T exts in Mathematis, v ol. 87, Springer-V erlag, New Y ork, 1982. [5℄ Jonathan Corni k and P eter H. Kropholler, Homolo gi al niteness  onditions for mo dules over str ongly gr oup-gr ade d rings , Math. Pro . Cam bridge Philos. So . 120 (1996), no. 1, 4354. [6℄ , On  omplete r esolutions , T op ology Appl. 78 (1997), no. 3, 235250. [7℄ , Homolo gi al niteness  onditions for mo dules over gr oup algebr as , J. London Math. So . (2) 58 (1998), no. 1, 4962. [8℄ Martin Hamilton, When is gr oup  ohomolo gy nitary? , (Preprin t, Univ ersit y of Glasgo w 2007). FINIT AR Y GR OUP COHOMOLOGY AND EILENBER GMA C LANE SP A CES 17 [9℄ Allen Hat her, A lgebr ai top olo gy , Cam bridge Univ ersit y Press, Cam bridge, 2002. [10℄ P eter H. Kropholler, On gr oups of typ e (FP) ∞ , J. Pure Appl. Algebra 90 (1993), no. 1, 5567. [11℄ , Hier ar hi al de  omp ositions, gener alize d Tate  ohomolo gy, and gr oups of typ e (FP) ∞ , Com binatorial and geometri group theory (Edin burgh, 1993), London Math. So . Leture Note Ser., v ol. 204, Cam bridge Univ. Press, Cam- bridge, 1995, pp. 190216. [12℄ , Mo dules p ossessing pr oje tive r esolutions of nite typ e , J. Algebra 216 (1999), no. 1, 4055. [13℄ , Gr oups with many nitary  ohomolo gy funtors , (Preprin t, Univ ersit y of Glasgo w 2007). [14℄ P eter H. Kropholler and Guido Mislin, Gr oups ating on nite-dimensional sp a es with nite stabilizers , Commen t. Math. Helv. 73 (1998), no. 1, 122 136. [15℄ P eter H. Kropholler and Olympia T alelli, On a pr op erty of fundamental gr oups of gr aphs of nite gr oups , J. Pure Appl. Algebra 74 (1991), no. 1, 5759. [16℄ Guido Mislin, T ate  ohomolo gy for arbitr ary gr oups via satel lites , T op ology Appl. 56 (1994), no. 3, 293300. [17℄ Charles A. W eib el, A n intr o dution to homolo gi al algebr a , Cam bridge Studies in A dv aned Mathematis, v ol. 38, Cam bridge Univ ersit y Press, Cam bridge, 1994. Ha usdorff Center f or Ma thema tis, Universit ä t Bonn, Land wir tshaft- skammer (Neuba u), Endeniher Allee 60, 53115 Bonn E-mail addr ess : hamiltonmath.u ni- bo nn .de