Rings of coinvariants and $p$-subgroups

Abstract:

Let $\varrho :G\hookrightarrow GL(n, \mathbb {F})$ be a faithful representation of a finite group $G$ over the field $\mathbb {F}$ and let $V \cong \mathbb {F}^n$ be an $\mathbb {F}(G)$-module. It has been shown by L. Smith that if $n=3$ and the order of $G$ is divisible by the positive characteristic $p$ of $\mathbb {F}$, then $\mathbb {F} [V]^G$ is Cohen-Macaulay. Under the condition $n=3$ we prove the following conjecture through this remarkable result: If $\mathbb {F} [V]_G$ is a Poincaré duality algebra, then $\mathbb {F} [V]_{\operatorname {Syl}_{p}(G)}$ is a complete intersection, where $\operatorname {Syl}_{p}(G)$ is a Sylow $p$-subgroup of $G$.