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Depth and cohomological connectivity in modular invariant theory
Authors:Peter Fleischmann  Gregor Kemper  R James Shank
Institution:Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, United Kingdom ; Zentrum Mathematik - M11, Technische Universität München, Boltzmannstr.~3, 85748 Garching, Germany ; Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, United Kingdom
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 $\min\left\{\dim_K(V^P) + 2,\dim_K(V)\right\}$.

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