Serre theorem in group cohomology

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

Let $p$ denote a fixed prime number and $G$ a pro-$p$-group, that is, an inverse limit of finite $p$-groups (cf. also $p$-group). Assume that $G$ is not an elementary Abelian $p$-group (i.e. it is not isomorphic to $(\textbf{Z}/p)^I$ for some indexing set $I$, where $\textbf{Z}/p$ is cyclic of order $p$). Then Serre's theorem asserts that there exist non-trivial $\mod p$ cohomology classes $v_1,...,v_k\in H^1(G,\textbf{Z}/p)$ such that the product $\beta(v_1)...\beta(v_k)=0$, where $\beta:H^1(G,\textbf{Z}/p)\to H^2(G,\textbf{Z}/p)$ is the Bockstein operation associated to the exact coefficient sequence $0\to\textbf{Z}/p\to\textbf{Z}/p^2\to\textbf{Z}/p\to 0$ (see [a9] and Cohomology operation). Note that for $p=2$ this is simply the squaring operation.

For a finite $p$-group, this can be made more explicit as follows. Each cohomology class $v_i$ corresponds to a (non-zero) homomorphism $\phi_i:G\to\textbf{Z}/p$ and hence an index-$p$ subgroup $G_i\subset G$. The class $\beta(v_i)\in\text{Ext}^{2_{\textbf{Z}/p[G]}}(\textbf{Z}/p,\textbf{Z}/p)$ can be represented as an extension class

\begin{equation}0\to\textbf{Z}/p\to\textbf{Z}/p[G/G_i]\to\textbf{Z}/p[G/G_i]\to\textbf{Z}/p\to0,\end{equation}

where $\textbf{Z}/p[G/G_i]$ denotes the usual permutation module obtained by induction. When concatenated together, one obtains a representation of the product, which is an element in $\text{Ext}^{2k_{\textbf{Z}/p[G]}}\:(\textbf{Z}/p,\textbf{Z}/p)$, as

\begin{equation}0\to\textbf{Z}/p\to\textbf{Z}/p[G/G_k]\to\textbf{Z}/p[G/G_k]\to...\to\\\to\textbf{Z}/p[G/G_1]\to\textbf{Z}/p[G/G_1]\to\textbf{Z}/p]\to0,\end{equation}

which the theorem asserts to be the trivial extension class.

The original application of Serre's result was for proving that if $G$ is a profinite group without elements of order $p$, then the $p$-cohomological dimension of $G$ is equal to the $p$-cohomological dimension of $U$ for any open subgroup $U\subset G$ (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 $\mod p$ cohomology of a finite group $G$ is equal to the rank of the largest elementary Abelian $p$-subgroup in $G$. More precisely, Serre's theorem can be used to verify that for a finite non-Abelian $p$-group $G$, the Krull dimension of $H^*(G,\textbf{Z}/p)$ (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) MR1317096 Zbl 0828.55008 Zbl 0820.20060
[a2]	D.J. Benson, "Representations and cohomology II: Cohomology of groups and modules" , Studies in Advanced Math. , 32 , Cambridge Univ. Press (1991) Zbl 0731.20001
[a3]	J.F. Carlson, "Modules and group algebras" , ETH Lect. Math. , Birkhäuser (1994) MR1428452 MR1393196 MR1342784 MR1338985 MR1159220 MR0621286 MR0613859 MR0528565 MR0491914 MR0472985 MR0554578 MR0419506 MR0364411 Zbl 0883.20006 Zbl 0837.20010 Zbl 0762.20021 Zbl 0484.20005
[a4]	L. Evens, "Cohomology of groups" , Oxford Univ. Press (1992) MR1144017 MR0574102 MR0153725 Zbl 0742.20050 Zbl 0122.02804
[a5]	D. Quillen, "The spectrum of an equivariant cohomology ring I–II" Ann. of Math. , 94 (1971) pp. 549–602 MR0298694 Zbl 0247.57013
[a6]	D. Quillen, B. Venkov, "Cohomology of finite groups and elementary Abelian subgroups" Topology , 11 (1972) pp. 317–318 MR0294506 Zbl 0245.18010
[a7]	J.-P. Serre, "Sur la dimension cohomologique des groupes profinis" Topology , 3 (1965) pp. 413–420 MR0180619 Zbl 0136.27402
[a8]	J.-P. Serre, "Cohomologie Galoisienne" , Lecture Notes in Mathematics , 5 , Springer (1994) (Edition: Fifth) MR1324577 Zbl 0812.12002
[a9]	E. Spanier, "Algebraic topology" , Springer (1989) MR1325242 Zbl 0810.55001 Zbl 0477.55001 Zbl 0222.55001 Zbl 0145.43303