Gregor Kemper
Universitaet Heidelberg

"On the Cohen-Macaulay Property of Modular Invariant Rings",
IWR-R-Preprint 97-38, Heidelberg 1997

Abstract:

  If V is a faithful module for a finite group G over a field of
  characteristic p > 0, then the ring of invariants need not be
  Cohen-Macaulay if p divides the order of G. In this article the
  cohomology of G is used to study the question of
  Cohen-Macaulayness of the invariant ring.

  Let R = S(V^*) be the polynomial ring on which G acts. Then the main
  result can be stated as follow: If H^r(G,R) is nonzero for an r > 0
  and if all sigma in G of order p have rank(sigma - 1) >= r+2, then
  the invariant ring R^G is not Cohen-Macaulay. A corollary is that if
  p divides the order of G, then the ring of vector invariants of
  sufficiently many copies of V is not Cohen-Macaulay. A further
  result is that if G is a p-group and R^G is Cohen-Macaulay, then G
  is a bireflection group, i.e., it is generated by elements sigma
  with rank(sigma-1) <= 2.