On the Depth of Modular Invariant Rings for the Groups
$C_p \times C_p$
Jonathan Elmer and Peter Fleischmann
Abstract
Let $G$ be a finite group, $k$ a field of
characteristic $p$ and $V$ a finite dimensional $kG$-module. Let
$R:= {\rm Sym}(V^*)$, the symmetric algebra over the dual space
$V^*$, with $G$ acting by graded algebra automorphisms. Then it is
known that the depth of the invariant ring $R^G$ is at least
$\min\{\dim(V), \dim(V^P)+cc_G(R)+1\}$. A module $V$ for which the
depth of $R^G$ attains this lower bound was called flat by
Fleischmann, Kemper and Shank \cite{FKS}. In this paper some of the
ideas in \cite{FKS} are further developed and applied to certain
representations of $C_p \times C_p$, generating many new examples of
flat modules. We introduce the useful notion of ``strongly flat''
modules, classifying them for the group $C_2 \times C_2$, as well as
determining the depth of $R^G$ for any indecomposable modular
representation of $C_2 \times C_2$.