Publication status: appeared in Trans. Amer. Math. Soc. 358 (2006), 1605-1617; arXiv: math.AC/0312302
Abstract: We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group G. By definition, these are G-actions on Laurent polynomial algebras that stabilize the multiplicative group consisting of all monomials in the variables. For the most part, we concentrate on the case where the base ring is Z. Our main result states that if G acts non-trivially and the invariant ring is Cohen-Macaulay then the abelianized isotropy groups Gmab of all monomials m are generated by bireflections and at least one Gmab is non-trivial. As an application, we prove the multiplicative version of Kemper's 3-copies conjecture.
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