Depth and Detection in Modular Invariant Theory

Jonathan Elmer

Abstract 

Let $G$ be a finite group acting linearly on a vector space
$V$ over a field of characteristic $p$ dividing the group order, and
let $R:= S(V^*)$. We study the $R^G$ modules $H^i(G,R)$, for $i \geq
0$ with $R^G$ itself as a special case. There are lower bounds for
$\dep_{R^G}(H^i(G,R))$ and for $\dep(R^G)$. We show that a certain
sufficient condition for their attainment (due to Fleischmann,
Kemper and Shank \cite{FKS1}) may be modified to give a condition
which is both necessary and sufficient. We apply our main result to
classify the representations of the Klein four-group for which
depth($R^G$) attains its lower bound, a process begun in \cite{EF}.
We also use our new condition to show that the if $G= P \times Q$,
with $P$ a $p$-group and $Q$ an abelian $p'$-group, then the depth
of $R^G$ attains its lower bound if and only if the depth of $R^P$
does so.