Depth and detection in modular invariant theory

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 0 with R G itself as a special case. There are lower bounds for depth R G (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 [P. Fleischmann, G. Kemper, R.J. Shank, Depth and cohomological connectivity in modular invariant theory, Trans. Amer. Math. Soc. 357 (2005) 3605-3621]) 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 [J. Elmer, P. Fleischmann, On the depth of modular invariant rings for the groups C p × C p , in: Proc. Symmetry and Space, Fields Institute, 2006, preprint, 2007]. We also use our new condition to show that 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.

On the depth of separating algebras for finite groups

“…Cohen-Macaulay as a module over itself). The problem of calculating depth for invariant rings in the modular case has been widely explored, and solved in only a few simple cases, for instance, when G is cyclic (Ellingsrud and Skjelbred 1980), p-nilpotent with cyclic Sylow-p-subgroup (Fleischmann et al 2005), and when G is the Klein four group (Elmer and Fleischmann 2010;Elmer 2009). The corresponding question of depth for separating algebras has not, to the best of our knowledge, been studied at all.…”