Cohen-Macaulay multigraded modules

TL;DR

This paper characterizes the Cohen-Macaulay property of multigraded modules over standard N^r-graded algebras using sheaf cohomology, extending prior work on multi-Rees algebras.

Contribution

It provides a new cohomological criterion for Cohen-Macaulayness of multigraded modules and applies it to multi-Rees modules, broadening understanding in this area.

Findings

Cohen-Macaulayness characterized by sheaf cohomology vanishing

Extension of criteria to multi-Rees modules

Broader conditions for Cohen-Macaulayness in multigraded settings

Abstract

Let S be a standard N^r-graded algebra over a local ring A, and let M be a finitely generated Z^r-graded S-module. 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 (also called Rees modification). Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.