Relative invariants, ideal classes and quasi-canonical modules of modular rings of invariants

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 (1982) [15].