On the Gorenstein property for modular invariants

Let G⊂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 SG(V) the ring of G-invariants. We prove here the following results:Theorem – Suppose that G contains no pseudo-reflection (of any kind).(1)If SG(V) is Gorenstein, then G⊂SL(V).(2)If G⊂SL(V) then the Cohen–Macaulay locus of SG(V) coincides with its Gorenstein locus. In particular if SG(V) is Cohen–Macaulay then it is also Gorenstein.This extends well-known results of K. Watanabe in case . It also confirms a special case of a conjecture due to G. Kemper, E. Körding, G. Malle, B.H. Matzat, D. Vogel and G. Wiese. A similar extension is given to D. Bensonʼs theorem about the Gorenstein property of (S(V)⊗ΛG(V)), the polynomial tensor exterior algebra invariants. Our proof uses non-commutative algebra methods in an essential way.