On the Cohen-Macaulay property of multiplicative invariants

Abstract:

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 ${\mathcal {G}}_m^{\operatorname {ab}}$ of all monomials $m$ are generated by the bireflections in $\mathcal {G}_m$ and at least one ${\mathcal {G}}_m^{\operatorname {ab}}$ is non-trivial. As an application, we prove the multiplicative version of Kemper’s $3$-copies conjecture.