Asymptotic vanishing of homogeneous components of multigraded modules and its applications

TL;DR

This paper establishes conditions under which finitely many homogeneous components' vanishing implies asymptotic vanishing in multigraded modules, with applications to algebraic properties of ideals.

Contribution

It introduces a new criterion linking finite component vanishing to asymptotic behavior in multigraded modules, extending previous results on ideal normality.

Findings

Condition for finite homogeneous component vanishing implies asymptotic vanishing

Application to multi-Rees algebras of ideals

Extended and improved results on normality of monomial ideals

Abstract

In this article, we give a condition on the vanishing of finitely many homogeneous components which must imply the asymptotic vanishing for multigraded modules. We apply our result to multi-Rees algebras of ideals. As a consequence, we obtain a result on normality of monomial ideals, which extends and improves the results of Reid-Roberts-Vitulli, Singla, and Sarkar-Verma.