On the Cohen-Macaulay property of multiplicative invariants

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 the ring of rational integers. Our main result states that if $G$ acts non-trivially and the invariant algebra is Cohen-Macaulay then the abelianized isotropy groups $G_m\mathrm{/}[G_m,G_m]$ of all monomials m are generated by bireflections and at least one $G_m\mathrm{/}[G_m,G_m]$ is non-trivial. As an application, we prove the multiplicative version of Kemper’s 3-copies conjecture.