Supporting degrees of multi-graded local cohomolgoy modules

TL;DR

This paper extends previous work on the support of local cohomology modules to the multi-graded setting, providing new insights into their structure and properties in a more complex grading context.

Contribution

It introduces multi-graded analogues of earlier results relating local cohomology modules and their supporting degrees, expanding the theoretical framework.

Findings

Provides restrictions on supporting degrees of multi-graded local cohomology modules.

Establishes connections between multi-graded local cohomology and the geometry of $ ext{Proj}(R)$.

Extends classical results to a multi-graded context.

Abstract

For a finitely generated graded module $M$ over a positively-graded commutative Noetherian ring $R$, the second author established in 1999 some restrictions, which can be formulated in terms of the Castelnuovo regularity of $M$ or the so-called $a^*$-invariant of $M$, on the supporting degrees of a graded-indecomposable graded-injective direct summand, with associated prime ideal containing the irrelevant ideal of $R$, of any term in the minimal graded-injective resolution of $M$. Earlier, in 1995, T. Marley had established connections between finitely graded local cohomology modules of $M$ and local behaviour of $M$ across $\Proj(R)$. The purpose of this paper is to present some multi-graded analogues of the above-mentioned work.