Depth and detection in modular invariant theory

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 denote S(V∗). We study the R^G modules H^i(G, R), for i ≥ 0 with R^G itself as a special case. There are lower bounds for depth of (H^i(G, R)) and for depth(R^G). We show that a certain sufficient condition for their attainment (due to Fleischmann, Kemper and Shank) 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. We also use our new condition to show that the if G = P × 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.

Available online 12 May 2009