The Depth of Invariant Rings and Cohomology

AbstractLet G be a finite group acting linearly on a vector space V over a field K of positive characteristic p and let P≤G be a Sylow p-subgroup. Ellingsrud and Skjelbred [Compositio Math.41 (1980), 233–244] proved the lower bound depth(K[VG)≤min{dim(VP)+2, dim(V)} for the depth of the invariant ring, with equality if G is a cyclic p-group. Let us call the pair (G,V) flat if equality holds in the above. In this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness. For G of order not divisible by p2 it ensues that (G,V) is flat if and only if H1(G,K[V])≠0 or dim(VP)+2≥dim(V). We obtain a formul a for the depth of the invariant ring in the case that G permutes a basis of V and has order not divisible by p2. In this situation (G,V) is usually not flat. Moreover, we introduce the notion of visible flatness of pairs (G,V) and prove that this implies flatness. For example, the groups SL2(q), SO3(q), SU3(q), Sz(q), and R(q) with many interesting representations in defining characteristic are visibly flat. In particular, if G=SL2(q) and V is the space of binary forms of degree n or a direct sum of such spaces, then (G,V) is flat for all q=pr with the exception of a finite number of primes p. Along the way, we obtain results about the Buchsbaum property of invariant rings and about the depth of the cohomology modules Hi(G,K[V]). We also determine the support of the positive cohomology H+(G,K[V]) as a module over K[V]G. In the Appendix, the visibly flat pairs (G,V) of a group with BN-pair and an irreducible representation are classified