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On the Cohen-Macaulay property of multiplicative invariants. (English) Zbl 1129.13005
Summary: We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $$\mathcal{G}$$. By definition, these are $$\mathcal{G}$$-actions on Laurent polynomial algebras $$\Bbbk[x_1^{\pm 1},\dots,x_n^{\pm 1}]$$ that stabilize the multiplicative group consisting of all monomials in the variables $$x_i$$. For the most part, we concentrate on the case where the base ring $$\Bbbk$$ is $$\mathbb{Z}$$. Our main result states that if $$\mathcal{G}$$ acts non-trivially and the invariant ring $$\mathbb{Z} [x_1^{\pm 1},\dots,x_n^{\pm 1}]^\mathcal{G}$$ is Cohen-Macaulay, then the abelianized isotropy groups $${\mathcal{G}}_m^{{ab}}$$ of all monomials $$m$$ are generated by the bireflections in $$\mathcal{G}_m$$ and at least one $${\mathcal{G}}_m^{{ab}}$$ is non-trivial. As an application, we prove the multiplicative version of Kemper’s $$3$$-copies conjecture.

##### MSC:
 13A50 Actions of groups on commutative rings; invariant theory 13C14 Cohen-Macaulay modules 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
GAP
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