Depth and cohomological connectivity in modular invariant theory

Abstract:

Let $G$ be a finite group acting linearly on a finite-dimensional vector space $V$ over a field $K$ of characteristic $p$. Assume that $p$ divides the order of $G$ so that $V$ is a modular representation and let $P$ be a Sylow $p$-subgroup for $G$. Define the cohomological connectivity of the symmetric algebra $S(V^*)$ to be the smallest positive integer $m$ such that $H^m(G,S(V^*))\not =0$. We show that $\min \left \{\dim _K(V^P) + m + 1,\dim _K(V)\right \}$ is a lower bound for the depth of $S(V^*)^G$. We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if $G$ is $p$-nilpotent and $P$ is cyclic, then, for any modular representation, the depth of $S(V^*)^G$ is $\min \left \{\dim _K(V^P) + 2,\dim _K(V)\right \}$.