aa r X i v : . [ m a t h . R T ] A ug FINITE GENERATION OF COHOMOLOGY OF FINITE GROUPS
RAPHA¨EL ROUQUIER
Abstract.
We give a proof of the finite generation of the cohomology ring of a finite p -groupover F p by reduction to the case of elementary abelian groups, based on Serre’s Theorem onproducts of Bocksteins. Definitions and basic properties
Let k = F p . Given G a finite group, we put H ∗ ( G ) = H ∗ ( G, k ). We refer to [Ev] for resultson group cohomology.Given A a ring and M an A -module, we say that M is finite over A if it is a finitely generated A -module.Let G be a finite group and L a subgroup of G . We have a restriction map res GL : H ∗ ( G ) → H ∗ ( L ). It gives H ∗ ( L ) the structure of an H ∗ ( G )-module.We denote by norm GL : H ∗ ( L ) → H ∗ ( G ) the norm map. If L is central in G , then we haveres GL norm GL ( ξ ) = ξ [ G : L ] for all ξ ∈ H ∗ ( L ).When L is normal in G , we denote by inf GG/L : H ∗ ( G/L ) → H ∗ ( G ) the inflation map.Let E be an elementary abelian p -group. The Bockstein H ( E ) → H ( E ) induces an injectivemorphism of algebras S ( H ( E )) ֒ → H ∗ ( E ). We denote by H ∗ pol ( E ) its image. Note that H ∗ ( E )is a finitely generated H ∗ pol ( E )-module and given ξ ∈ H ∗ ( E ), we have ξ p ∈ H ∗ pol ( E ).2. Finite generation for finite groups
The following result is classical. We provide here a proof independent of the finite generationof cohomology rings.
Lemma 2.1.
Let G be a p -group and E an elementary abelian subgroup. Then, H ∗ ( E ) is finiteover H even ( G ) .Proof. The result is straightforward when G is elementary abelian. As a consequence, given G ,it is enough to prove the lemma when E is a maximal elementary abelian subgroup. We provethe lemma by induction on | G | . Let Z ≤ Z ( G ) with | Z | = p . Let P be a complement to Z in E .Let A = inf EE/Z ( H ∗ pol ( E/Z )). Let x be a generator of H ( Z ) ∼ → H ( E/P ) and y = inf EE/P ( x ).We have H ∗ pol ( E ) = A ⊗ k [ y ] . Let ξ = res GE (norm GZ ( x ) p ). We have res EZ ( ξ ) = x p [ G : Z ] , so ξ − y p [ G : Z ] ∈ H ∗ pol ( E ) ∩ ker res EZ = A > H ∗ pol ( E ). We deduce that H ∗ ( E ) is finite over its subalgebra generated by A and ξ . Date : October 2014.The author was partially supported by the NSF grant DMS-1161999.
By induction, H ∗ ( E/Z ) is finite over H ∗ ( G/Z ). We deduce that H ∗ ( E ) is finite over itssubalgebra generated by ξ and inf EE/Z res
G/ZE/Z H ∗ ( G/Z ) = res GE inf GG/Z H ∗ ( G/Z ). (cid:3) Let us recall a form of Serre’s Theorem on product of Bocksteins [Se]. We state the resultover the integers for a useful consequence stated in Corollary 2.3.
Theorem 2.2 (Serre) . Let G be a finite p -group. Assume G is not elementary abelian. Then,there is n ≥ , there are subgroups H , . . . , H n of index p of G and an exact sequence of Z G -modules → Z → Ind GH n Z → · · · → Ind GH Z → Z → defining a zero class in Ext n Z G ( Z , Z ) .Proof. Serre shows there are elements z , . . . , z m ∈ H ( G, Z /p ) such that β ( z ) · · · β ( z m ) = 0.The element z i corresponds to a surjective morphism G → Z /p with kernel H i , and we identifyInd GH i Z with Z [ G/H i ] = Z [ σ ] / ( σ p − σ is a generator of G/H i . The element β ( z i ) ∈ H ( G, Z /p ) is the image of the class c i ∈ H ( G, Z ) given by the exact sequence0 → Z σ + ··· + σ p − −−−−−−−−→ Ind GH i Z − σ −−→ Ind GH i Z augmentation −−−−−−−→ Z → . Let c = c · · · c m ∈ H m ( G, Z ). The image of c in H m ( G, Z /p ) vanishes, hence c ∈ pH m ( G, Z ).Fix r such that | G | = p r . Since | G | H > ( G, Z ) = 0, we deduce that c r = 0. (cid:3) We will only need the case R = F p of the corollary below. We denote by D b ( RG ) the derivedcategory of bounded complexes of finitely generated RG -modules. Corollary 2.3.
Let G be a finite group and R a discrete valuation ring with residue field ofcharacteristic p or a field of charactetistic p . Assume x p − = 1 has p − solutions in R .Let I be the thick subcategory of D b ( RG ) generated by modules of the form Ind GE M , where E runs over elementary abelian subgroups of G and M runs over one-dimensional representationsof E over R .We have I = D b ( RG ) .Proof. Assume first G is an elementary abelian p -group. Let L be a finitely generated RG -module. Consider a projective cover f : P → L and let L ′ = ker f . The R -module L ′ is free, so L ′ is an extension of RG -modules that are free of rank 1 as R -modules. So L ′ ∈ I and similarly P ∈ I , hence L ∈ I . As a consequence, the corollary holds for G elementary abelian.Assume now G is a p -group that is not elementary abelian. We proceed by induction on | G | .Let L be a finitely generated RG -module. By induction, Ind GH Res GH ( L ) ∈ I whenever H is aproper subgroup of G . Applying L ⊗ Z G − to the exact sequence of Theorem 2.2, we obtain anexact sequence 0 → L → Ind GH n Res GH n ( L ) → · · · → Ind GH Res GH ( L ) → L → . Since that sequence defines the zero class in Ext n ( L, L ), it follows that L is a direct summandof 0 → Ind GH n Res GH n ( L ) → · · · → Ind GH Res GH ( L ) → D b ( RG ). We deduce that L ∈ I .Finally, assume G is a finite group. Let P be a Sylow p -subgroup of G and let L a finitelygenerated RG -module. We know that Ind GP Res GP ( L ) ∈ I . Since L is a direct summand ofInd GP Res GP ( L ), we deduce that L ∈ I . (cid:3) INITE GENERATION OF COHOMOLOGY OF FINITE GROUPS 3
Remark 2.4.
Corollary 2.3 implies a corresponding generation result for the stable categoryof RG . That was observed by the author in the mid 90s and communicated to J. Carlson whowrote an account of this in [Ca]. Theorem 2.5 (Golod, Venkov, Evens) . Let G be a finite p -group. The ring H ∗ ( G ) is finitelygenerated. Given M a finitely generated kG -module, then H ∗ ( G, M ) is a finitely generated H ∗ ( G ) -module. Note that the case where G is an arbitrary finite group follows easily, cf [Ev]. Proof.
Let S be a finitely generated subalgebra of H even ( G ) such that H ∗ ( E ) is a finitelygenerated S -module for every elementary abelian subgroup E of G . Such an algebra exists byLemma 2.1.Let J be the full subcategory of D b ( kG ) of complexes C such that the S -module H ∗ ( G, C ) = L i Hom D b ( kG ) ( k, C [ i ]) is finitely generated.Let C → C → C be a distinguished triangle in D b ( kG ). We have a long exact sequence · · · → H i ( C ) → H i ( C ) → H i ( C ) → H i +1 ( C ) → · · · Assume C , C ∈ J . Let I be a finite generating set of H ∗ ( C ) as an S -module and J a finitegenerating set of ker( H ∗ ( C ) → H ∗ +1 ( C )) as an S -module. Let I ′ be the image of I in H ∗ ( C )and let J ′ be a finite subset of H ∗ ( C ) with image J . Then, I ′ ∪ J ′ generates H ∗ ( C ) as an S -module, hence C ∈ J .Note that if C ⊕ C ′ ∈ J , then C ∈ J . We deduce that J is a thick subcategory of D b ( kG ).Let E be an elementary abelian subgroup of G . Since H ∗ ( G, Ind GE ( k )) ≃ H ∗ ( E, k ) is a finitelygenerated S -module, we deduce that Ind GE ( k ) ∈ J .We deduce from Corollary 2.3 that J = D b ( kG ). (cid:3) References [Ca] J. Carlson,
Cohomology and induction from elementary abelian subgroups , Q. J. Math. (2000), 169–181.[Ev] L. Evens, “The cohomology of groups”, Oxford University Press, 1991.[Se] J.P. Serre, Sur la dimension cohomologique des groupes profinis , Topology (1965), 413-420. Department of Mathematics, UCLA, Box 951555, Los Angeles, CA 90095-1555, USA
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