Decomposition of graded local cohomology tables

TL;DR

This paper characterizes the structure of the cone formed by local cohomology tables of certain graded modules over polynomial rings, providing a way to decompose any such table into extremal components.

Contribution

It identifies the extremal rays and facets of the cone of local cohomology tables for modules of dimension at most two and offers algorithms for their decomposition.

Findings

Describes extremal rays and facets of the cone.

Shows any point in the cone can be decomposed into extremal points.

Provides algorithms for decomposition.

Abstract

Let $R$ be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded $R$-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide algorithms to obtain decompositions in terms of extremal points and facets.