Associated Primes for Cohomology Modules

Let k be a field of finite characteristic p, and G a finite group acting on the left on a finite dimensional k-vector space V . Then the dual vector space V ∗ is naturally a right kG-module, and the symmetric algebra of the dual, R := Sym(V ∗), is a polynomial ring over k on which G acts naturally by graded algebra automorphisms, and if k is algebraically closed can be regarded as the space k[V ] of polynomial functions on V . The G-fixed points of R under this action form a ring, which we denote by R and call the ring of invariants. If k is algebraically closed, R can be regarded as the set of G-invariant polynomial functions on V , or the ring of coordinate functions on the quotient space V/G. The ring of invariants R is the central object of study in invariant theory. The situation becomes modular when we assume p divides the order of G. Let P be a (fixed) Sylow-p-subgroup of G. Since the ring of invariantsR coincides with the zeroth cohomologyH(G,R), we can regard R as the zeroth degree part of the cohomology ring H∗(G,R), and as such, the higher cohomology modules H(G,R) become R-modules via the cup product. One can often learn more about the structure of modular rings of invariants by studying these higher cohomology modules; for example, in [3] Ellingsrud and Skjelbred showed that H(G,R) is Cohen-Macaulay for G cyclic of order p. They then used this result to find a formula for the depth the ring of invariants R in this case. This approach was also used in [7], [9] and [10] to answer questions about the depth or Cohen-Macaulay property of modular invariant rings. If X < G, we may define a mapping TrX : R X → R as follows: let S be a set of right coset representatives of X in G. Then we define