Relative Invariants, Ideal Classes and Quasi-Canonical Modules of Modular Rings of Invariants

TL;DR

This paper introduces quasi canonical modules for modular invariant rings, providing a criterion for Gorenstein properties and classifying reflexive modules using class groups and semi-invariants, generalizing previous results.

Contribution

It generalizes classical and recent results on invariant rings by defining quasi canonical modules and establishing a Gorenstein criterion in the modular setting.

Findings

Defined quasi canonical modules for modular invariant rings.

Derived a quasi Gorenstein criterion using 1-cocycles.

Classified reflexive rank one modules via class groups and semi-invariants.

Abstract

We describe "quasi canonical modules" for modular invariant rings $R$ of finite group actions on factorial Gorenstein domains. From this we derive a general "quasi Gorenstein criterion" in terms of certain 1-cocycles. This generalizes a recent result of A. Braun for linear group actions on polynomial rings, which itself generalizes a classical result of Watanabe for non-modular invariant rings. We use an explicit classification of all reflexive rank one $R$-modules, which is given in terms of the class group of $R$, or in terms of $R$-semi-invariants. This result is implicitly contained in a paper of Nakajima (\cite{Nakajima:rel_inv}).