# Serre theorem in group cohomology

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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].

How to Cite This Entry:
Serre theorem in group cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serre_theorem_in_group_cohomology&oldid=51526
This article was adapted from an original article by Alejandro Adem (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article