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

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

2010-11-23

arxiv.org/abs/1011.5153v1

Mathematics

Commutative Algebra

16 pages

Scientific paper

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}).

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