On the Gorenstein property for modular invariants
Amiram Braun
Dept of Mathematics, University of Haifa, Haifa, Israel 31905
Abstract. Let G \subset GL(V) be a finite group, where V is a finite
dimensional vector space over a field F of arbitrary characteristic.
Let S(V) be the symmetric algebra of V and S(V)^G the ring of
G-invariants. We prove here the following results:
Theorem. Suppose that G contains no pseudo-reflection (of any kind).
(1) If S(V)^G is Gorenstein, then G \subset SL(V).
(2) If G \subset SL(V) then the Cohen-Macaulay locus of S(V)^G
coincides with its Gorenstein locus. In particular if S(V)^G is
Cohen-Macaulay then it is also Gorenstein.
This extends well known results of K. Watanabe in case (char F,|G|)=1.
A similar extension is given to a theorem of D. Benson. Our proof
uses non-commutative algebra methods in an essential way.