Finite generation of Tate cohomology

FINITE GENERA TION OF T A TE COH OMOLOGY JON F. CARLSON, SUNIL K. CHEBOLU, AND J ´ AN MIN ´ A ˇ C De dic ate d to Pr ofessor Luchezar Avr amov on his sixtie th bi rt hday. Abstract. Let G be a finite group and l et k b e a field of c haracteristic p . Giv en a finitely generated indecomposable non-pro jectiv e k G -mo dule M , we conjecture that if the T ate cohomology ˆ H ∗ ( G, M ) of G with co efficien ts in M is finitely generated o ver the T ate cohomology ring ˆ H ∗ ( G, k ), then the supp ort v ariet y V G ( M ) of M i s equal to the entire maximal ideal sp ectrum V G ( k ). W e pr o ve v arious results which support this conjecture. The conv erse of this conjecture is established for mo dules in the connect ed component of k in the s table Auslander-Reiten quiver f or k G , but it is shown to be f alse i n general. It is also shown that all finitely generated k G - modules ov er a group G hav e finitely generated T ate cohomology i f and only if G has p erio dic cohomology . 1. Introduction T ate co ho mology was intro duced by T ate in his celebrated pap er [14] where he prov ed the main theorem of class field theory in a remark ably simple w ay using T ate cohomolo gy . After C a rtan and Eilenberg’s treatment [9] o f T a te c o homology and Swan’s basic results on free g r oup a c tions on spheres [13], T ate cohomo logy b ecame one of the basic to ols in c ur rent ma thematics. Our aim in this pap er is to address a fundamental question: when is the T a te cohomo logy with co efficients in a mo dule finitely generated over the T ate c ohomology r ing of the gro up. Suppo se G be a finite group and le t k be a field o f characteristic p . If M is a finitely generated k G -mo dule, then a well-known result in gr oup co homology due to Golo d, Evens and V enko v says that H ∗ ( G, M ) is finitely generated as a gr aded mo dule ov er H ∗ ( G, k ). Our goal is to inv estigate a similar finite-genera tion result for T ate cohomolog y . Mo re precisely , if M is a finitely ge ne r ated k G -mo dule, then we wan t to know whether the T ate cohomolo gy ˆ H ∗ ( G, M ) of G with co efficients in M is finitely generated as a gra ded mo dule over the T a te cohomolo gy ring ˆ H ∗ ( G, k ). In Section 2 we explain one reas o n for b eing in teres ted in this problem. In g eneral, it seems that the T ate co homology of a mo dule is seldom finitely generated, which is a str ik ing co nt ra st to the situation with ordinar y cohomolo gy . How ever, there are some notable exceptions. Our inv estiga tions have le d us to a conjecture which we state as follows. Date : August 18, 2021. 2000 Mathematics Subje ct Classific ation. Primary 20C20, 20J06; Secondary 55P42. Key wor ds and phr ases. T ate cohomology , finite generation, p erio dic modules, supp ort v arieties, stable mo dule category , alm ost split sequence. The firs t author is partially supp orted by a grant from NSF and the third author is supported from NSERC. 2 JON F. CARLSON, SUNIL K. CHEBOLU, AND J ´ AN MIN ´ A ˇ C Conjecture 1.1. L et G b e a finite gr oup and let M b e an inde c omp osab le finitely gen- er ate d k G - mo dule such that H ∗ ( G, M ) 6 = { 0 } . If ˆ H ∗ ( G, M ) is finitely gener ate d over ˆ H ∗ ( G, k ) , then t he supp ort variety V G ( M ) of M is e qual to t he entir e max imal ide al sp e ctr u m V G ( k ) of the gr oup c ohomolo gy ring. The condition that H ∗ ( G, M ) 6 = 0 is certainly necess ary since there a r e many mo dules with prop er s upp or t v arieties and v anishing co homology [5]. Perhaps it is necessa ry to require that M lie s in the thick sub catego r y of the s table c ategory g enerated by k . W e have evidence for the conjecture from tw o directions. First, the results of [3] indicate that pro ducts in nega tive T a te cohomolo gy ar e o ften zero a nd we can use this to develop b oundedness co nditions on finitely gener ated mo dules ov er T a te cohomolo gy . Under the right cir c umstances, these conditions imply infinite gener ation o f the T ate cohomolog y . Secondly , for g roups having p - rank at least tw o, we can show that many per io dic mo dules fail to have finitely generated T a te co homology . Indeed, we prove tha t for any such group there is a t least one mo dule whose T a te cohomolog y is not finitely generated. Hence, the only groups having the prop erty that every finitely generated k G -mo dule has finitely genera ted T ate coho mology hav e p -ra nk one or zer o. On the other ha nd, in g eneral, there are numerous mo dules which have finitely gener - ated T ate coho mology . In the last s ection we s how so me wa ys in which these mo dules ca n be constr ucted. It turns out that the constructions a re cons is ten t with the Auslander - Reiten quiver for k G -mo dules. That is, if a nonpro jectiv e mo dule in a connected co m- po nent o f the Auslander-Reiten quiver has finitely gener ated T ate co homology , then s o do es every mo dule in that comp onent. The pap er is o rganized as follows. W e b egin in Section 2 by explaining how we had naturally arr ived at the pro blem of finite g e ne r ation o f T a te cohomolog y . Sections 3 and 4 deal with mo dules who se T a te cohomolog y is not finitely generated and contain pro ofs in the tw o directions mentioned ab ov e. In Se c tio n 5 we prove a ffirmative r e sults which pr ovides a g o o d s ource of mo dules whose T ate cohomo logy is finitely generated. Throughout the pap er G denotes a non- trivial finite gr oup, a nd a ll k G - mo dules ar e assumed to be finitely g enerated. W e use standa rd facts and notation of the s table mo dule ca tegory of k G [7], supp ort v arieties [2, 8], and of a lmost split sequences [2]. 2. Universal ghosts in stmo d ( k G ) Here we explain br iefly how we had arrived at the proble m of finite genera tion of T a te cohomolog y . Mor e deta ils can b e found in [10, 11]. The questio n of finite generation is very natur a l. The finite ge ne r ation of the ordinary cohomolog y has been very imp ortant in the development of the theory of supp ort v arieties a nd in o ther connections. F or T ate cohomolog y , almost nothing is known ab out the the ques tion of finite generation b eyond what is in this pap er. The following natura l question was raised in [11]: when do es the T ate cohomo logy functor detect trivial maps in the stable mo dule categ ory stmo d( k G ) of finitely gene r ated k G -mo dules? A map φ : M → N be t ween finitely g enerated k G -mo dules is said to b e a ghost if the induced map in T ate cohomolog y g roups Hom kG (Ω i k , M ) − → Hom kG (Ω i k , N ) is zer o for each integer i . With this definition, the ab ove ques tion is equiv alent to ask ing when every ghost ma p in s tmod ( k G ) triv ial. In a ddressing this question, it is conv enient FINITE GENE RA TION OF T A TE COHOMOLOGY 3 to hav e a universal g ho st out of any finitely generated k G -mo dule M in stmo d( k G ), i.e., a ghost map φ : M → N in stmo d( k G ) such that e very g host out o f M fac to rs thro ugh N via φ . The p oint is tha t if the universal g host v anishes, then all gho sts v anish. So the pr o blem b oils down to finding a universal g host out of M (if it exists) for every mo dule M in stmo d( k G ). The p oint is that if the T ate co ho mology ˆ H ∗ ( G, M ) is finitely g enerated as a graded mo dule over ˆ H ∗ ( G, k ), then a universal ghost out of M can b e constr ucted in stmo d( k G ). This is done as follows. Let { v j } b e a finite set of homogeneous ge nerators for ˆ H ∗ ( G, M ) as a ˆ H ∗ ( G, k )-mo dule. These g enerators ca n b e assembled into a map M j Ω | v j | k − → M in stmo d ( k G ). This map can then b e co mpleted to a tr iangle (2.1) M j Ω | v j | k − → M Ψ M − → F M . By construc tio n, it is clear that the fir st map in the ab ove triang le is surjective on the functors Hom kG (Ω l k , − ) for each l . Therefore, the second map Ψ M m ust b e a ghost. Thu s we hav e the following prop os itio n. Prop ositio n 2.1 . Supp ose that M is a finitely gener ate d k G -mo dule such t hat ˆ H ∗ ( G, M ) is finitely gener ate d as a gr ade d mo dule over ˆ H ∗ ( G, k ) . Then the map Ψ M : M → F M in the ab ove triangle is a universal ghost out of M . Pr o of. Universality of Ψ M is easy to s ee. F or the last statement, we note that b e c ause the sum is finite, ⊕ j Ω | v j | k is finitely gener ated.  3. Modules with bounds on finitel y genera ted submo dules In this s ection we apply our main metho d for showing that mo dules over T ate coho- mology are not finitely g enerated. W e ex plore the implications of the fo llowing condition. The materia l in this section draws heavily on the metho ds in tro duced in the pap er [3]. Definition 3.1 . W e say that a gr aded mo dule T = ⊕ n ∈ Z T n ov er ˆ H ∗ ( G, k ) has b ounded finitely genera ted submo dules if for any m there is a num b er N = N ( m ) such that the submo dule S of T generated by ⊕ n>m T n is contained in ⊕ n>N T n . Lemma 3.2. If a gr ade d m o dule T = ⊕ n ∈ Z T n over ˆ H ∗ ( G, k ) has b ounde d finitely gen- er ate d s u bmo dules and if T n 6 = { 0 } for arbitr arily smal l (me aning ne gative) values of n , then T is not a finitely gener ate d mo dule over ˆ H ∗ ( G, k ) . Pr o of. The pro of is an immediate consequence of the definition. The p oint is that any finitely g enerated submo dule of T is contained in P n>m T n for some m a nd hence cannot generate all of T .  R emark 3.3 . There is a mor e genera l formulation of the b oundedness condition that might b e useful, though we do not use it in this pap er. W e say that T ∗ has submax imal growth o f finitely generated submo dules if the degre e of the p ole a t 1 of the Poincar´ e series for the submo dule S of T ge ne r ated by ⊕ n>m T n is strictly smalle r than the degree 4 JON F. CARLSON, SUNIL K. CHEBOLU, AND J ´ AN MIN ´ A ˇ C of the p ole at 1 of the Poincar´ e series of T . The Poincar´ e series for T is the Laurent series f T ( t ) = ∞ X n = −∞ (Dim( T n )) t n . Its p ole at t = 1 is a mea sure o f the growth ra te of T in nega tiv e degr ees. That is , if the po le has degree d , then there is a num be r c such that Dim( T − n ) ≤ cn d − 1 for a ll n , while for any constant c there ex ists a natur al num ber n such that Dim( T − n ) > cn d − 2 . It is straightforward to show that any T ∗ which has s ubmaximal growth of finitely g enerated submo dule is not finitely gener ated ov er ˆ H ∗ ( G, k ). The gr aded mo dules ov er the T a te cohomo logy ring that we are in teres ted in have the form ˆ H ∗ ( G, L ), w her e L is a k G -mo dule. W e r emind the reader that if ˆ H i ( G, L ) 6 = 0 for some i , then it is a ls o non-z e ro for infinitely ma n y negative and infinitely many p ositive v alues of i [5, Thm. 1.1]. Mor eov er, a standard arg umen t shows that any no n-pro jective mo dule L in the thick sub categ ory gener a ted by k ha s non-v anishing T a te co homology . Here, the thick sub category genera ted by k is the sma llest full sub categ o ry of stmo d( k G ) that co nt ains k and is closed under exact triangles and direc t summands. W e use the next lemma several times in wha t follows. Lemma 3. 4. Su pp ose that we have an exact se quenc e E : 0 / / L / / M / / N / / 0 of k G -mo dules wher e E r epr esents an element ζ in Ex t 1 kG ( N , L ) . Cup pr o duct with the element ζ induc es a homomorphism ζ : ˆ H ∗ ( G, N ) − → ˆ H ∗ ( G, L )[1] . L et K ∗ b e t he kernel of the mult iplic ation by ζ , and let J ∗ b e the c okernel of multiplic atio n by ζ . Then we have an exact se quenc e of ˆ H ∗ ( G, k ) -mo dules 0 / / J ∗ / / ˆ H ∗ ( G, M ) / / K ∗ / / 0 . Mor e over, if K ∗ is not finitely gener ate d over ˆ H ∗ ( G, k ) , then neither is ˆ H ∗ ( G, M ) . Pr o of. The pro of is a straightforward consequence of the natura lity of the long exact sequence on T ate cohomolog y . That is, we hav e a sequence . . . ζ / / ˆ H n ( G, L ) / / ˆ H n ( G, M ) / / ˆ H n ( G, N ) ζ / / ˆ H n +1 ( G, L ) / / . . . and we note that the collection of the maps ζ in the long exact sequence is a map of degree 1 o f ˆ H ∗ ( G, k )-mo dules ζ : ˆ H ∗ ( G, N ) / / ˆ H ∗ ( G, L )[1] . (The symbol X [ i ] indicates the shift of the ˆ H ∗ ( G, k )-mo dule X by i degrees.) The last statement is a conseq uence of the fac t that quotient mo dules o f finitely generated mo dules are finitely genera ted.  Now supp ose that ζ ∈ H d ( G, k ) for d > 0 a nd that ζ 6 = 0 . W e have an exa c t sequence 0 / / L ζ / / Ω d k ζ / / k / / 0 FINITE GENE RA TION OF T A TE COHOMOLOGY 5 where ζ in the sequence is a homomo rphism (uniquely) repr e s ent ing the cohomo logy element ζ . In the corresp o nding long exact sequence on T ate cohomology . . . ζ / / ˆ H n − 1 ( G, k ) / / ˆ H n ( G, L ζ ) / / ˆ H n ( G, Ω d k ) ζ / / ˆ H n ( G, k ) / / . . . , the homomorphism lab eled ζ is multiplication by ζ . That is, it is degree d map: ζ : ˆ H ∗ ( G, k )[ − d ] − → ˆ H ∗ ( G, k ) . Here we a re using the fact that ˆ H s ( G, Ω d k ) ∼ = ˆ H s − d ( G, k ). As a result, we have, as in Lemma 3.4, a n exa ct sequence of ˆ H ∗ ( G, k )-mo dules 0 / / J ∗ [ − 1] / / ˆ H ∗ ( G, L ζ ) / / K ∗ [ − d ] / / 0 . where J ∗ and K ∗ are the cokernel and kernel of multiplication b y ζ , resp ectively . Lemma 3. 5. Su pp ose that ζ ∈ H d ( G, k ) is a re gular element on H ∗ ( G, k ) . Then (1) K m = { 0 } for al l m ≥ 0 , and (2) J m = { 0 } for al l m < 0 . Pr o of. The firs t sta tement is the definition that ζ is a reg ular element in H ∗ ( G, k ). The second s tatement is a conseq uence of Lemma 3.5 o f [3]. F o r the sake of completeness we inc lude a pro of. F o r t > 0, let h , i : ˆ H − t − 1 ( G, k ) ⊗ ˆ H t ( G, k ) / / ˆ H − 1 ( G, k ) ∼ = k be the T ate duality . Let ζ 1 , . . . , ζ s be a k -basis for ˆ H − m − 1 ( G, k ). Then b ecause multi- plication by ζ , ˆ H − m − 1 ( G, k ) / / ˆ H − m + d − 1 ( G, k ) is a monomor phism (since − m − 1 ≥ 0), the ele ments ζ ζ 1 , . . . , ζ ζ s are linea rly indep en- dent . So there must e xist elements γ 1 , . . . , γ s in ˆ H m − d ( G, k ) such that for all i a nd j , we have h γ i , ζ ζ j i = h γ i ζ , ζ j i = δ i,j where b y δ i,j we mean the usual Kr oneck er delta. A consequence of this is that the elements γ 1 ζ , . . . , γ s ζ must be line a rly independent a nd hence must form a bas is for ˆ H m ( G, k ). This proves the lemma.  There are many examples of groups for which all pro ducts in neg ative c o homology are ze r o. F o r example we r emind the reader of the following theor em fr o m [3 ]. Theorem 3. 6. Supp ose t hat the or dinary c ohomo lo gy ring H ∗ ( G, k ) has a r e gular s e- quenc e of length 2. Then the pr o duct of any two elements in ne gative c ohomol o gy is zer o. In p articular, this happ ens whenever t he p -r ank of the c enter of a Sylow p - sub gr oup of G is at le ast 2. The seco nd statement of the theo rem was pr ov ed by Duflot (see Theo rem 12.3 .3 of [8]). 6 JON F. CARLSON, SUNIL K. CHEBOLU, AND J ´ AN MIN ´ A ˇ C Prop ositio n 3.7. Supp ose that G has p -r ank at le ast two and that ˆ H ∗ ( G, k ) has the pr op erty that t he pr o duct of any t wo elements in ne gative de gr e es is zer o. If ζ ∈ H d ( G, k ) ( d > 0 ) is a r e gular element for H ∗ ( G, k ) , t hen ˆ H ∗ ( G, L ζ ) is not fi nitely gener ate d as a mo dule over ˆ H ∗ ( G, k ) . Pr o of. As b efore, let K ∗ be the kernel of the multiplication by ζ on ˆ H ∗ ( G, k ). The fact that K ∗ is not zero in infinitely many negative degrees follows easily fr o m Lemma 2.1 of [3] and the fact that there is no bo und on the dimensions of the spaces ˆ H n ( G, k ) for negative v alues of n . W e have shown that K ∗ has elements o nly in negative degrees and pro ducts of element s in nega tive degr ees are zero. Therefore , K has bo unded finitely generated submo dules and by L e mma 3.2 it is not finitely g enerated. Then by Lemma 3.4 neither is ˆ H ∗ ( G, L ζ ).  Example 3.8 . W e consider the Klein four group G = V 4 . The classificatio n o f the indecomp osable k V 4 -mo dules over a field k of characteristic 2 is well-known; se e [2, V ol. 1, Thm. 4.3.2] for instance. If the field k is alg ebraically clo sed then every even dimensional indeco mpo sable no n-pro jective mo dule ha s the form L ζ m for some ζ ∈ H 1 ( H, k ). O n the o ther hand, every indecomp os a ble mo dule of o dd dimension is isomorphic to Ω i k for some i . Because every no nz e r o e le ment of H 1( G, k ) is regula r , we hav e that for any indec o mpo sable k G -mo dule M , the T ate cohomolog y of M is finitely generated ov er ˆ H ∗ ( G, k ) if and only if V G ( M ) = V G ( k ). In particular, Conjecture 1 .1 holds in this case. A t this p oint we need to r ecall a technical no tion. W e say that a cohomolo g y element ζ ∈ H n ( G, k ) annihila tes the cohomolog y of a mo dule M , if the cup pro duct with ζ is the zero op er ator on Ext ∗ kG ( N , M ) for all mo dules N . The element ζ annihila tes the cohomolog y of M if and o nly if L ζ ⊗ M ∼ = Ω n M ⊕ Ω M ⊕ P wher e P is some pro jective mo dule. See Section 9.7 of [8]. F rom the s a me source we hav e that if p > 2 and if ζ ∈ H ∗ ( G, k ) with n even, then ζ annihilates the coho mology of L ζ . Even in the case tha t p = 2, we k now that the degree one element s cor resp onding to maximal subgr o ups of a 2 -gro up hav e the prop erty that ζ annihilates the coho mology of L ζ . Mor e ov er, the pr o duct of any tw o elements with this prop erty has this pr op erty . W e are now prepar ed to prove the ma in theor em o f this section. Theorem 3.9. S u pp ose that ˆ H ∗ ( G, k ) has the pr op erty that the pr o duct of any two elements in ne gative de gr e es is zer o. L et ζ ∈ H ∗ ( G, k ) b e a r e gular element of de gr e e d . In the c ase that p = 2 , assume t hat ζ annihilates the c ohomolo gy of L ζ . If M is a finitely gener ate d k G -mo dule such that ˆ H ∗ ( G, M ) 6 = 0 and V G ( M ) ⊆ V G h ζ i , t hen ˆ H ∗ ( G, M ) is not fi n itely gener ate d as an ˆ H ∗ ( G, k ) -mo dule. Pr o of. Since ˆ H ∗ ( G, M ) 6 = 0, by Lemma 3 .2 it is enough to show that ˆ H ∗ ( G, M ) has bo unded finitely genera ted s ubmo dules . Beca us e of the condition tha t V G ( M ) ⊆ V G h ζ i , we k now that some p ow er of ζ , say ζ t , annihilates the coho mology o f M . Hence it follows that L ζ t ⊗ M ∼ = Ω M ⊕ Ω td M ⊕ P, for some pro jectiv e mo dule P . Thus, ˆ H ∗ ( G, M ) has bo unded finitely generated sub- mo dules if a nd only if ˆ H ∗ ( G, L ζ t ⊗ M ) also has this pr o p e rty . Note that if p > 2, then FINITE GENE RA TION OF T A TE COHOMOLOGY 7 the degree o f ζ must b e even b ecause ζ is reg ular and hence not nilp otent. So for any v alue of p we hav e that ζ annihilates the cohomo lo gy of L ζ . The action of ˆ H ∗ ( G, k ) on ˆ H ∗ ( G, L ζ t ⊗ M ) facto rs through the map ˆ H ∗ ( G, k ) − → d Ext ∗ kG ( L ζ t , L ζ t ) ∼ = H ∗ ( G, ( L ζ t ) ∗ ⊗ L ζ t ) ∼ = H ∗ ( G, Ω − dt L ζ t ⊕ Ω − 1 L ζ t ), since for any ζ of degree d we have that L ∗ ζ ∼ = Ω − d − 1 L ζ (see [8], Section 11 .3). So the tar get of that map has bo unded finitely gener a ted submo dules. Now let m b e any integer. Without lo ss of g e nerality we ca n assume that m < 0. Let M = M n ≥ m ˆ H n ( G, L ζ t ⊗ M ) ⊆   M n ≥ m d Ext n kG ( L ζ t , L ζ t )     M n ≥ m ˆ H n ( G, L ζ t ⊗ M )   . F rom Definition 3.1, we know that there exists a n umber N such that ˆ H ∗ ( G, k ) · M n ≥ m d Ext n kG ( L ζ t , L ζ t ) ⊆ M n ≥ N d Ext n kG ( L ζ t , L ζ t ) . Hence, we hav e that ˆ H ∗ ( G, k ) · M ⊆ ˆ H ∗ ( G, k ) ·   M n ≥ m d Ext n kG ( L ζ t , L ζ t )     M n ≥ m ˆ H n ( G, L ζ t ⊗ M )   ⊆   M n ≥ N d Ext n kG ( L ζ t , L ζ t )     M n ≥ m ˆ H n ( G, L ζ t ⊗ M )   ⊆ M n ≥ N + m ˆ H n ( G, L ζ t ⊗ M ) . Therefore, ˆ H n ( G, L ζ t ⊗ M ) has b ounded finitely gener ated s ubmo dules.  Using the r esults of the theorem, w e can settle Co njectur e 1.1 in some sp ecial cases as in the fo llowing. Corollary 3.10. L et p > 2 . Su pp ose that the gr oup G has an ab elian Sylow p -sub gr oup with p -r ank at le ast two. If M is a finitely gener ate d k G -mo dule with H ∗ ( G, M ) 6 = 0 and if V G ( M ) is a pr op er su bvariety of V G ( k ) , t hen ˆ H ∗ ( G, M ) is n ot finitely gener ate d as a mo dule over ˆ H ∗ ( G, k ) . Pr o of. If V G ( M ) is a prop er subv ar iety of V G ( k ), then V G ( M ) ⊆ V G ( ζ ) for s ome non- nilpo ten t element ζ ∈ H ∗ ( G, k ). But be cause the Sylow subgr oup of G is an a belia n p -gro up, every no n-nilpo tent element in H ∗ ( G, k ) is reg ular, a nd mo r eov er, any tw o elements in negative degrees in ˆ H ∗ ( G, k ) hav e zero pro duct. So the pr o of is complete by the previous theorem.  R emark 3.11 . W e should note tha t ˆ H ∗ ( G, M ) having infinitely genera ted T a te cohomol- ogy do es no t require that it hav e b ounded finitely genera ted submo dules or even sub- maximal g rowth of finitely gener ated submo dule (see 3 .3). F or an example, co nsider the semidihedral 2-gr oup G of order 1 6 a nd let k = F 2 . Let M = L ζ , where ζ ∈ H 1( G, F 2 ) is a nonnilp otent element. See the example in Section 4 of [3]. Then it can be s een tha t M ∼ = Ω k ↑ G H where H is the subgr oup defined by the class ζ , that is, the maximal sub- group of G on which ζ v anishes. So we s ee that M ⊗ M ∼ = Ω M ⊕ Ω M ⊕ ( k G ) 12 . Hence, we can see by the results of the next sectio n, that ˆ H ∗ ( G, M ) is not finitely generated as 8 JON F. CARLSON, SUNIL K. CHEBOLU, AND J ´ AN MIN ´ A ˇ C a mo dule ov er ˆ H ∗ ( G, k ). O n the other hand, ζ is not a regular ele ment, so the mo dule K ∗ , which is the kernel of ζ , do es not hav e b ounded finitely gener a ted submo dules or submaximal growth of finitely gener ated submo dules . How ever, a careful analysis shows that K ∗ is no t finitely genera ted. W e end this section by s howing that there is a c ounterexample to the conv erse of our conjecture 1.1. W e sus p ect that such examples are numerous. W e give only an outline of the pro of in one example, leaving the details to the reader. Prop ositio n 3.12 . Ther e exists a mo dule M with V G ( M ) = V G ( k ) such that ˆ H ∗ ( G, M ) is not fin itely gener ate d over ˆ H ∗ ( G, k ) . Sketch of Pr o of. Let G = h x, y i b e an elementary ab elian g roup o f or der p 2. Here k ha s characteristic p . W e a ssume tha t p > 2 . Let H = h y i and let L = k ↑ G H be the induced mo dule. The mo dule of our exa mple is the extensio n M in the non-split exact sequence E : 0 / / k σ / / M / / L / / 0 . The mo dule M ca n b e descr ibed by gene r ators and r elations as the quotient o f k G b y the ideal gene r ated by ( y − 1)2 and ( x − 1)( y − 1 ). The map σ sends 1 to y − 1. Note that bec ause the dimensio n of M is rela tively prime to p , we must hav e that V G ( M ) = V G ( k ). W e hav e a s equence 0 → J ∗ → ˆ H ∗ ( G, M ) → K ∗ → 0 as in 3.4, whe r e J ∗ and K ∗ are, resp ectively , the co kernel and kernel of the map ˆ H ∗ ( G, L ) − → ˆ H ∗ +1 ( G, k ) given by multiplying b y the clas s of E . Our int ere s t is in the submo dule K ∗ ⊆ ˆ H ∗ ( G, L ). Because L = k ↑ G H , we have by the Eckmann-Shapiro Lemma tha t ˆ H ∗ ( G, L ) ∼ = ˆ H ∗ ( h y i , k ). Consequently , ˆ H ∗ ( G, L ) has dimensio n one in every degree a nd the action of ˆ H ∗ ( G, k ) on ˆ H ∗ ( G, L ) factor s throug h the res triction map ˆ H ∗ ( G, k ) − → ˆ H ∗ ( h y i , k ), which we know is the zero map in negative degr ees. Therefor e K ∗ has b ounded finitely ge ne r ated submo dules. So b y Lemmas 3.2 and 3.4, the pro of is co mplete when we show that K ∗ is no t zero in infinitely many negative deg rees. W e take the lo ng exact sequence in cohomolo g y corres po nding to the dual E ∗ of the exact s equence E , noting that the mo dule L is self dual. The connecting homo morphism is cup pro duct with the class of the sequence E ∗ . By Eckmann-Shapiro, it is the restric- tion map follow ed by cup pr o duct with a nonzero class η in H 1 ( H , k ). Since η 2 = 0 , we hav e that the image has dimens io n one, if n is even a nd δ is the zero map if n is o dd. Hence b eca use Dim H n ( G, k ) = n + 1, we must also have that H n ( G, M ∗ ) also has dimension n + 1 . By T a te duality , H − n ( G, M ) is dual to H n − 1 ( G, M ∗ ) fo r n > 0. Ther efore H − n ( G, M ) has dimensio n n , which is the sa me as the dimension of H − n ( G, k ). Returning to the long exa ct sequence corresp o nding to E , we arg ue by dimensions that the connec ting homomorphism is the ze r o map in e very second degree. So we show that the dimensio n of K n is zer o if n is negative a nd even and is one other wise. This completes the pro o f.  4. Periodic modul es In this s ection, we present o ur second piece of ev ide nc e for the Conjecture 1.1. W e show that for any group G with p -r a nk at le a st 2, ther e is a finitely genera ted mo dule M with the prop erty that ˆ H ∗ ( G, End k M ) is not finitely gener ated as a ˆ H ∗ ( G, k )-mo dule. FINITE GENE RA TION OF T A TE COHOMOLOGY 9 W e recall tha t a finite g r oup G has p erio dic cohomo logy , meaning that the tr ivial mo dule k is per io dic, if and o nly if G ha s p -rank zero o r one (see [2] or [8]). Theorem 4.1. Supp ose that t he gr oup G has p -r ank at le ast 2. L et M b e a n on- pr oje ctive p erio dic k G -mo dule su ch that H ∗ ( G, M ) 6 = 0 . Then ˆ H ∗ ( G, Hom k ( M , M )) ∼ = d Ext ∗ kG ( M , M ) is not finitely gener ate d as a ˆ H ∗ ( G, k ) -mo dule. Thus for any finite gr oup G such that p divides the or der of G , the T ate c ohomolo gy of every finitely gener ate d k G -mo dules is finitely gener ate d over ˆ H ∗ ( G, k ) if and only if G has p -r ank one, me aning that t he Sylow p -sub gr oup of G is either a cyclic gr oup or a gener alize d Quaternion gr oup. Pr o of. Let E = h x 1 , . . . , x n i b e a maximal e lement ar y ab elian p - s ubgroup such that the restriction M E is not a free mo dule. Ther e exists a n element α = ( α 1 , . . . , α n ) ∈ k n and a corresp o nding c y clic shifted subgro up h u α i , u α = 1 + n X i =1 α i ( x i − 1) such that the restriction of M to h u α i is not pro jectiv e (see Section 5 .8 of [2]). Hence, the ident ity homomorphis m Id M : M − → M do es no t factor throug h a pro jectiv e k h u α i - mo dule. As a consequence , the ma p k − → Hom k ( M , M ) which sends 1 ∈ k to Id M m ust represent a non-zero class in ˆ H 0( h u α i , Hom k ( M , M )). The next thing that we no te is that the r estriction map res G, h u α i : ˆ H d ( G, k ) − → ˆ H d ( h u α i , k ) is the zero map if d < 0. The reaso n is that the r estriction ma p res E , h u α i : ˆ H d ( E , k ) − → ˆ H d ( h u α i , k ) is z e ro by [3] since E has rank at lea st 2 . Now supp ose tha t M is p erio dic of p erio d t . F or every m we hav e that Ω mt M ∼ = M and ther e e x ists an element ζ m ∈ d Ext mt kG ( M , M ) ∼ = ˆ H mt ( G, Hom k ( M , M )) such that ζ m is no t zero on restr ic tion to h u α i . Tha t is, ζ m is repres ent ed by a co cycle k − → Hom k ( M , M ) ∼ = Ω mt Hom k ( M , M ) which do e s no t factor through a pro jectiv e mo dule on res triction to h u α i . Suppo se that ˆ H ∗ ( G, Hom k ( M , M )) is finitely gener ated as a mo dule over ˆ H ∗ ( G, k ). Then ther e exist generato rs µ 1 , . . . , µ r of ˆ H ∗ ( G, Hom k ( M , M )), having degrees d 1 , . . . , d r , resp ectively . Choos e an integer m such that mt < min { d i } . W e must hav e that ζ m = P r i =1 γ i µ i for so me γ i ∈ ˆ H mt − d i ( G, k ). But now, fo r every i , we have that mt − d i is neg ative. Hence res G, h u α i ( γ i ) = 0 for every i . Therefor e, s ince restriction onto a shifted subgr oup is a homomor phis m we hav e that r es G, h u α i ( ζ m ) = 0. But this is a co ntradiction. T o prov e the la s t statement of the theor em, we rec a ll that every finite group with non-trivial Sy low p -subgr oup admits a finitely g enerated non-pr o jective and p erio dic k G -mo dule in the thick sub category g e nerated by k . If the gro up has p -rank one, then k is such a mo dule. If the p -rank of G is gre ater than one, then any tensor pro duct L ζ 1 ⊗ · · · ⊗ L ζ n is p erio dic and is in the thick s ub ca tegory gener ated by k , provided the 10 JON F. CARLSON, SUNIL K. CHEBOLU, AND J ´ AN MIN ´ A ˇ C dimension of the v ariety V G ( ζ 1 ) ∩ · · · ∩ V G ( L ζ n ) has dimensio n o ne (see Chapter 10 of [8] or Chapter 5 of [2], V olume 2).  There is o ne other concept which ties up well with finite generatio n o f T ate cohomo l- ogy , and this is a ghost pro jective clas s in the stmo d( k G ). Consider the pair ( P , G ), where P is a class of o b jects is o morphic in stmo d ( k G ) to finite dire c t s ums of susp en- sions of k , and G is a cla s s of all ghosts in stmo d( k G ). Rec all that a ghost is a map of k G -mo dules that is zero in T ate cohomo logy . W e say that ( P , G ) is a gho st pro jectiv e class if the following 3 conditions are sa tisfied. (1) The class of all maps X → Y such that the comp osite P → X → Y is zero for all P in P and all maps P → X is precisely G . (2) The cla ss of all ob jects P such that the comp osite P → X → Y is zero for all maps X → Y in G and all maps P → X is precisely P . (3) F or ea ch o b ject X there is a n ex a ct tr iangle P → X → Y with P in P and X → Y in G . The first question that comes to mind is whether the g host pro jective class exists in stmo d( k G ). W e answer this in the next theor em. Theorem 4.2 . F or G a finite gr oup, such that p divides the or der of G . The ghost pr oje ctive class exists in stmo d( k G ) if and only if G has p -r ank one. Pr o of. It is clear fro m the definition of a gho st that P and G are orthogona l, i.e., the comp osite P → M h → N is zero for a ll P in P , for all h in G , and all maps P → M . So by [12, Le mma 3.2] it r e mains to show that for a ll finitely g e ne r ated k G -mo dules M , there exists a triangle P → M → N such that P is in P a nd M → N is in G . The exact tria ngle (2.1) has this for m in the case that the T ate coho mo logy o f M is finitely generated ov er ˆ H ∗ ( G, k ). F or the conv ers e, suppo se that M is a finitely ge nerated k G -mo dule. Since the gho st pro jective class exists, we hav e an exact triangle M Ω i k ⊕ θ i − → M ρ − → N in stmo d( k G ) where ρ is a gho st. W e claim that the finite set { θ i } genera te ˆ H ∗ ( G, M ) as a mo dule over ˆ H ∗ ( G, k ). T o se e this, c o nsider any ele ment γ in ˆ H t ( G, M ) repre sented by a co cycle γ : Ω t k → M . Since ρ is a g host, we get the following commutativ e diagra m: L Ω i k ⊕ θ i / / M ρ / / N Ω t k ⊕ r i c c γ O O ργ =0 > > } } } } } } } } F rom this diagr am, we infer tha t γ = P r i θ i . This shows that ˆ H ∗ ( G, M ) is finitely generated, as desired. Hence, by Theorem 4.1, the p - rank o f G is one.  5. Modules with finitel y genera ted T a te cohomology It is clea r that any mo dule M which is a direct sum of Heller translates Ω n k has finitely g enerated T ate coho mology . This is simply b ecause ˆ H ∗ ( G, M ) is a direct sum of co pies o f ˆ H ∗ ( G, k ) which hav e b een suitably tr anslated in degr ees. Also any finitely FINITE GENE RA TION OF T A TE COHOMOLOGY 11 generated mo dules over a group with per io dic coho mology has finitely gener ated T ate cohomolog y . In this section we show that in genera l there are many mor e mo dules with this prop erty . Notice that every one of the mo dules which we discuss has the pro p e rty that V G ( M ) = V G ( k ), consistent with Conjecture 1.1 . W e firs t cons ider the T ate co ho mology of mo dules M which can o ccur a s the middle term o f an exact sequence of the for m 0 / / Ω m k / / M / / Ω n k / / 0 for some v alues of m a nd n . Such a s equence represents an element ζ in Ext 1 kG (Ω n k , Ω m k ) ∼ = d Ext n +1 − m kG ( k , k ) ∼ = ˆ H n +1 − m ( G, k ) . F or the purp oses of exa mining the T ate co homology o f M there is no lo ss of genera lit y in applying the shift o per ator Ω − m . Conseq uent ly we can a ssume that the sequence has the form (5.1) 0 / / k / / M / / Ω n k / / 0 for some n and that ζ ∈ ˆ H n +1 ( G, k ). The principal result of this s ection is the fo llowing. Theorem 5.1. Supp ose that the c ohomolo gy of G is n ot p erio dic and that for the mo dule M and c ohomolo gy element ζ as ab ove, the m ap ζ : ˆ H ∗ ( G, k ) / / ˆ H ∗ ( G, k ) given by mult iplic ation by ζ has a finite dimensional image. Then the T ate c ohomolo gy ˆ H ∗ ( G, M ) is finitely gener ate d as a mo dule over ˆ H ∗ ( G, k ) . There ar e many example of s equences sa tisfying the conditions of the theor em. In particular, it is often the c a se that multiplication b y an element ζ in neg ative cohomolo gy will have finite dimensiona l image. An example is the element in degree − 1 which represents the almo st s plit seq uence fo r the mo dule k . Details of this example are given below. In a dditio n, if the depth of H ∗ ( G, k ) is tw o or mor e then a ll pro ducts inv olving elements in negative degr ees are zero, and the principal idea l gener ated b y a ny element in negative coho mology co ntains no non-zero elements in p ositive degrees (see [3 ]). Hence, m ultiplication by a ny element ζ in nega tive c o homology ha s finite dimensional image. Pr o of. As in Lemma 3.4, we hav e an exact sequence of ˆ H ∗ ( G, k )-mo dules 0 / / J ∗ / / ˆ H ∗ ( G, M ) / / K ∗ [ − n ] / / 0 , where K ∗ is the kernel of mu ltiplication by ζ o n ˆ H ∗ ( G, k ) and J ∗ is the cokernel. By assumption, the image of multiplication by ζ has finite total dimension. This means that in all but a finite num ber o f degrees r , m ultiplication by ζ is the zero map. Clea rly , J ∗ is finitely g enerated over ˆ H ∗ ( G, k ). So, ˆ H ∗ ( G, M ) is finitely genera ted as a mo dule ov er ˆ H( G, k ) if and only if K ∗ has the same prop erty . First we v ie w K ∗ as a mo dule ov er the o rdinary cohomolo gy ring H ∗ ( G, k ). The elements in no n-negative deg r ees form a submo dule M ∗ = P i ≥ 0 K i , which is finitely generated over H ∗ ( G, k ). Let N ∗ be the ˆ H ∗ ( G, k )-submo dule mo dule of K ∗ generated by M ∗ . Our ob jective is to show that N ∗ = K ∗ , thereby proving the finite gener ation 12 JON F. CARLSON, SUNIL K. CHEBOLU, AND J ´ AN MIN ´ A ˇ C of K ∗ . W e notice first that K n ⊆ N ∗ for n ≥ 0. It r emains o nly to s how the same for n < 0 . Because the quotient of ˆ H ∗ ( G, k ) b y K ∗ is finite dimens io nal, we must have that ˆ H n ( G, k ) = K n for n sufficie ntly large. F or some s ufficien tly lar ge n , we can find an element γ in K n which is a re gular element for the or dina ry cohomolo gy ring H ∗ ( G, k ). F or example, by Duflot’s Theorem (see 3.6), a ny element whos e restriction to the center of a Sylow p -subgr oup of G is not nilp otent will serve this purp ose (see [2] or [8]). L e t θ be the image of γ in K n . W e know that θ is not zero . W e also know that multiplication by γ is a surjective map γ : ˆ H m − n ( G, k ) / / ˆ H m ( G, k ) whenever m < 0 (see Lemma 3.5 of [3]). Hence, for any m < 0, we must hav e that ˆ H m − n ( G, k ) θ = K m . Since θ ∈ N ∗ , we get that K m ⊆ N ∗ for a ll m < 0. Hence, K ∗ = N ∗ is finitely gener ated as a mo dule over ˆ H ∗ ( G, k ).  One application of the theor em is the following. Corollary 5.2 . The midd le term of t he almost split se quenc e 0 / / Ω 2 k σ / / M / / k / / 0 ending with k has finitely gener ate d T ate c ohomolo gy. Pr o of. If G has p -rank zer o o r one, then by Theorem 4.1, all mo dules have finitely generated T ate coho mology . So we as sume that G has p -r ank a t least 2. T he almost split seque nc e corres po nds to an elemen t ζ in ˆ H − 1 ( G, k ). One of the defining pro per ty of the almost s plit sequence is that for any mo dule N , the co nnec ting ho momorphism δ in the co rresp onding sequence . . . / / Hom kG ( N , M ) σ ∗ / / Hom kG ( N , k ) δ / / Ext 1 kG ( N , Ω 2 k ) / / . . . is non-zero if a nd only if N ∼ = k . This c onnecting homomor phism is mult iplicatio n by ζ . No w any element γ in ˆ H d ( G, k ) is r epresented by a map γ : Ω − d k → k . Hence , we see that ζ γ = 0 whenever d 6 = 0 . Therefore, multiplication by ζ on ˆ H ∗ ( G, k ) has finite-dimensional image.  Prop ositio n 5. 3. L et N b e a finitely gener ate d inde c omp osable non-pr oje ct ive k G - mo dule that is not isomorphic to Ω i k for any i . Consider the almost split se quenc e 0 / / Ω2 N / / M / / N / / 0 ending in N . If N has finitely gener ate d T ate c ohomolo gy, then so do es the midd le t erm M . Pr o of. In a similar way as in the previo us pro of, for any i , the connec ting homomorphis m δ in the sequence . . . / / Hom kG (Ω i k , M ) / / Hom kG (Ω i k , N ) δ / / Ext 1 kG (Ω i k , Ω 2 N ) / / . . . is zer o b eca use Ω i k 6 ∼ = N . As a conse q uence, δ induces the ze ro map on T ate co homology . Hence, the lo ng exact sequence in T ate co homology br e a ks into short exact sequences: 0 / / ˆ H ∗ ( G, Ω2 N ) / / ˆ H ∗ ( G, M ) / / ˆ H ∗ ( G, N ) / / 0 FINITE GENE RA TION OF T A TE COHOMOLOGY 13 It is now clear that if N has finitely gener ated T ate co homology , then s o do e s M .  In summary , combining the las t tw o results we hav e the following theorem. Theorem 5.4 . L et C b e a c onne cte d c omp onent of t he stable Auslander-R eiten quiver asso cia te d t o k G . Then either al l mo dules in C have finitely gener ate d T ate c ohomolo gy or no m o dule in C has this pr op erty. Mor e over, al l mo dules in the c onne cte d c omp onent of the quiver which c ontains k have finitely gener ate d T ate c ohomolo gy. It is shown in [1, Pr o p o sition 5.2] that all mo dules M in the connected co mp onent o f the quiver which contains k have the pro per ty V G ( M ) = V G ( k ). Thus the last theore m is c o nsistent with Conjecture 1.1 Ac kno wledgme n ts: The first author is gr ateful to the R WTH in Aa chen for their hospitality and the Alexa nder von Humboldt F oundation for fina ncial supp ort during a v is it to Aachen dur ing which parts of this pap er were written. The fir st and second authors had some fruitful co nversations on this work at MSRI during the spring 20 08 semester on Repr e sentation theory and related topics. They b oth w ould like to thank MSRI for its hospitality . References [1] M. Ausl ander and J. F. Carlson. 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CHEBOLU, AND J ´ AN MIN ´ A ˇ C Dep ar tment of Ma thema tics, University of Georgia, A then s, GA 30602, USA E-mail addr e ss : jfc@mat h.uga.edu Dep ar tment of Mathema tics, Illinois St a te University, Campus box 45 20, Norm al, IL 61790, USA E-mail addr e ss : schebol @ilstu.edu Dep ar tment of Mat hematic s, University of Western Ont ario, London, ON N6 A 5B7, Canada E-mail addr e ss : minac@u wo.ca