Serre theorem in group cohomology

A theorem proved by J.-P. Serre in 1965 about the cohomology of pro--groups which has important consequences in group cohomology and representation theory (cf. also Pro--group; Cohomology of groups). The original proof appeared in [a7], a proof in the context of finite group cohomology appears in [a1].

Let denote a fixed prime number and a pro--group, that is, an inverse limit of finite -groups (cf. also -group). Assume that is not an elementary Abelian -group (i.e. it is not isomorphic to for some indexing set , where is cyclic of order ). Then Serre's theorem asserts that there exist non-trivial cohomology classes such that the product , where is the Bockstein operation associated to the exact coefficient sequence (see [a9] and Cohomology operation). Note that for this is simply the squaring operation.

For a finite -group, this can be made more explicit as follows. Each cohomology class corresponds to a (non-zero) homomorphism and hence an index- subgroup . The class can be represented as an extension class

where denotes the usual permutation module obtained by induction. When concatenated together, one obtains a representation of the product, which is an element in , as

which the theorem asserts to be the trivial extension class.

The original application of Serre's result was for proving that if is a profinite group without elements of order , then the -cohomological dimension of is equal to the -cohomological dimension of for any open subgroup (see [a8] for more on this; cf. also Cohomological dimension).

However, it is also a basic technical result used in proving the landmark result (see [a5] and [a6]) that the Krull dimension (cf. Dimension) of the cohomology of a finite group is equal to the rank of the largest elementary Abelian -subgroup in . More precisely, Serre's theorem can be used to verify that for a finite non-Abelian -group , the Krull dimension of (the maximal rank of a polynomial subalgebra) is determined on maximal proper subgroups, hence leading to an inductive argument which can be reduced to elementary Abelian subgroups.

This, in turn, can be extended to arbitrary finite groups and to cohomology with coefficients in a modular representation. Indeed, it is a basic result in the theory of complexity and cohomological varieties in representation theory. This is explained [a2], [a3] and [a4].

References

[a1]	A. Adem, R.J. Milgram, "Cohomology of finite groups" , Grundlehren , 309 , Springer (1994)
[a2]	D.J. Benson, "Representations and cohomology II: Cohomology of groups and modules" , Studies in Advanced Math. , 32 , Cambridge Univ. Press (1991)
[a3]	J.F. Carlson, "Modules and group algebras" , ETH Lect. Math. , Birkhäuser (1994)
[a4]	L. Evens, "Cohomology of groups" , Oxford Univ. Press (1992)
[a5]	D. Quillen, "The spectrum of an equivariant cohomology ring I–II" Ann. of Math. , 94 (1971) pp. 549–602
[a6]	D. Quillen, B. Venkov, "Cohomology of finite groups and elementary Abelian subgroups" Topology , 11 (1972) pp. 317–318
[a7]	J.-P. Serre, "Sur la dimension cohomologique des groupes profinis" Topology , 3 (1965) pp. 413–420
[a8]	J.-P. Serre, "Cohomologie Galoisienne" , Lecture Notes in Mathematics , 5 , Springer (1994) (Edition: Fifth)
[a9]	E. Spanier, "Algebraic topology" , Springer (1989)