## Abstract

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 S(V)^{G} the ring of G-invariants. We prove here the following results:. TheoremSuppose that G contains no pseudo-reflection (of any kind).(1)If S(V)^{G} is Gorenstein, then G⊂SL(V).(2)If G⊂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 (charF,|G|)=1. 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. Benson's theorem about the Gorenstein property of (S(V)⊗Λ(V))^{G}, the polynomial tensor exterior algebra invariants. Our proof uses non-commutative algebra methods in an essential way.

Original language | English |
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Pages (from-to) | 81-99 |

Number of pages | 19 |

Journal | Journal of Algebra |

Volume | 345 |

Issue number | 1 |

DOIs | |

State | Published - 1 Nov 2011 |

## Keywords

- Gorenstein rings
- Modular invariants

## ASJC Scopus subject areas

- Algebra and Number Theory