Bass numbers of local cohomology of cover ideals of graphs

TL;DR

This paper introduces splitting techniques to compute Bass numbers and Lyubeznik numbers of local cohomology modules for cover ideals of various graph classes, simplifying the analysis of their algebraic properties.

Contribution

It provides a new method to explicitly compute Bass numbers and describe the injective resolutions of local cohomology modules for broad classes of graphs.

Findings

Explicit formulas for Bass numbers of cover ideals of graphs.

A simple criterion for vanishing of local cohomology modules based on graph connectivity.

Computation of the shape of injective resolutions for these modules.

Abstract

We develop splitting techniques to study Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally we are able to compute all the Bass numbers and the shape of the injective resolution of local cohomology modules by considering the connected components of the corresponding subgraphs. Indeed our method gives us a very simple criterion for the vanishing of these local cohomology modules in terms of the connected components.