Lyubeznik numbers of monomial ideals

TL;DR

This paper investigates Lyubeznik numbers associated with squarefree monomial ideals, establishing a connection with minimal free resolutions and providing bounds on the injective dimension of local cohomology modules.

Contribution

It introduces a novel interpretation of Lyubeznik numbers via the acyclicity of linear strands in Alexander dual ideals, linking local cohomology to free resolutions.

Findings

Lyubeznik numbers as obstructions to acyclicity

A new dictionary between local cohomology and free resolutions

Bounds on injective dimension based on small support

Abstract

We study Bass numbers of local cohomology modules supported on squarefree monomial ideals paying special attention to Lyubeznik numbers. We build a dictionary between local cohomology modules and minimal free resolutions that allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of the Alexander dual ideals. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support.