On the Cohen-Macaulay property of modular invariant rings

If V is a faithful module for a finite group G over a field of characterist ic p, then the ring of invariants need not be Cohen-Macaulay if p divides t he order of G. In this article the cohomology of G is used to study the que stion of Cohen-Macaulayness of the invariant ring. One of the results is a classification of all groups for which the invariant ring with respect to t he regular representation is Cohen-Macaulay. Moreover; it is proved that if p divides the order of G, then the ring of vector invariants of sufficient ly many copies of V is not Cohen-Macaulay. A further result is that if: G i s a p-group and the invariant ring is Cohen-Macaulay, then G is a bireflect ion group, i.e., it is generated by elements which fix a subspace of V of c odimension at most 2. (C) 1999 Academic-Press.