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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)
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