Cohen–Macaulay multigraded modules

Abstract

Let $S$ be a standard $\mathbb{N}^r$-graded algebra over a local ring $A$, and let $M$ be a finitely generated $\mathbb{Z}^r$-graded module over $S$. We characterize the Cohen–Macaulayness of $M$ in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen–Macaulayness of multi-Rees modules. Our work extends previous studies on the Cohen–Macaulayness of multi-Rees algebras.