Equivariant cohomology and depth

Abstract

Let $n \geq 1$ be an integer, let $V=(\mathbb{Z}/2\mathbb{Z})^{n}$ and let $X$ be a $V$-CW-complex. If $X$ is a finite $CW$-complexe, the equivariant modulo $2$ cohomology of the $V$-CW-complexe $X$, denoted by $H_{V}^{*}(X, \mathbb{F}_{2})$, is a finite type module over the modulo $2$ cohomology of the group $V$, denoted by $H^{*}(V, \mathbb{F}_{2})$. Let $dth_{H^{*}V}H_{V}^{*}(X, \mathbb{F}_{2})$ be the depth of the finite type $H^{*}(V, \mathbb{F}_{2})$-module $H_{V}^{*}(X, \mathbb{F}_{2})$ relatively to the augmentation ideal, $\widetilde{H^{*}}(V, \mathbb{F}_{2})$, of $H^*(V, \mathbb{F}_{2})$. \medskip\\ The aim of this paper is to prove the following result: \medskip\\ {\bf Theorem}: For every subgroup $W$ of $V$, we have: $dth_{H^{*}W}H_{W}^{*}(X, \mathbb{F}_{2}) \leq dth_{H^{*}V} H_{V}^{*}(X, \mathbb{F}_{2}) $.