The Cohen - Macaulay Property of Invariant Rings

If V is a faithful module for a nite group G over a eld 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) 6 = 0 for an r > 0 and if all 2 G of order p have rank(? 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 suuciently 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 bireeection group, i.e., it is generated by elements with rank(? 1) 2.