On the Depth of Modular Invariant Rings for the Groups Cp x Cp

Let G be a finite group, k a field of characteristic p and V a finite dimen-

sional kG-module. Let R :=Sym(V?), the symmetric algebra over the dual spaceV?,

with G acting by graded algebra automorphisms. Then it is known that the depth of

the invariant ring RG is at least min{dim(V),dim(VP)+ccG(R)+1}. A module V

for which the depth of RG attains this lower bound was called flat by Fleischmann,

Kemper and Shank [13]. In this paper some of the ideas in [13] are further developed

and applied to certain representations of Cp ×Cp, generating many new examples

of flat modules. We introduce the useful notion of “strongly flat” modules, classi-

fying them for the group C2 ×C2, as well as determining the depth of RG for any

indecomposable modular representation of C2×C2.