Cohomological properties of non-standard multigraded modules

TL;DR

This paper investigates cohomological properties of non-standard multigraded modules, focusing on invariance under Veronese transforms and establishing vanishing theorems that imply asymptotic constancy of depth.

Contribution

It introduces new invariance results for generalized depth and proves vanishing theorems for local cohomology in the multigraded setting.

Findings

Generalized depth is invariant under Veronese transforms

Vanishing theorems for local cohomology modules

Depth of Veronese modules stabilizes asymptotically

Abstract

In this paper we study some cohomological properties of non-standard multigraded modules and Veronese transforms of them. Among others numerical characters, we study the generalized depth of a module and we see that it is invariant by taking a Veronese transform. We prove some vanishing theorems for the local cohomology modules of a multigraded module; as a corollary of these results we get that the depth of a Veronese module is asymptotically constant.