On the depth of modular invariant rings for the groups C_p x C_p

Let G be a finite group, k a field of characteristic p and V a finite dimensional kG-module. Let R denote 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 [13]. In this paper some of the ideas in [13] are further developed and applied to certain representations of C_p × C_p, generating many new examples of flat modules. We introduce the useful notion of “strongly flat” modules, classi-fying them for the group C_2 × C_2, as well as determining the depth of R^G for any indecomposable modular representation of C_2 × C_2.

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