Equivariant cohomology and depth

Abstract

Let $n\ge 1$ 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 $H_V^{\ast }(X,{\mathbb{F}}_2)$, is a finite type module over the modulo $2$ cohomology of the group $V$, denoted by $H^{\ast }(V,{\mathbb{F}}_2)$. Let $dt{h}_{H^{\ast }V}{H}_V^{\ast }(X,{\mathbb{F}}_2)$ be the depth of the finite type $H^{\ast }(V,{\mathbb{F}}_2)$-module $H_V^{\ast }(X,{\mathbb{F}}_2)$ relatively to the augmentation ideal, $\widetilde{H^{\ast }}(V,{\mathbb{F}}_2)$, of $H^{\ast }(V,{\mathbb{F}}_2)$. \medskip\\ The aim of this paper is to prove the following result: \medskip\\ {\bf Theorem}: For every subgroup $W$ of $V$, we have: $dt{h}_{H^{\ast }W}{H}_W^{\ast }(X,{\mathbb{F}}_2)\le dt{h}_{H^{\ast }V}{H}_V^{\ast }(X,{\mathbb{F}}_2)$.

16 pages