An Eisenbud-Goto type inequality for Stanley-Reisner ideals and simplicial complexes

TL;DR

This paper establishes an upper bound on the Leray number of simplicial complexes based on facet attachment, leading to an Eisenbud-Goto type inequality for Stanley-Reisner ideals, generalizing previous results.

Contribution

It introduces a new bound on the Leray number related to facet attachment and characterizes complexes achieving equality, extending Eisenbud-Goto inequalities to square-free monomial ideals.

Findings

Derived an upper bound on the Leray number based on facet attachment.

Characterized the structure of complexes where the bound is tight.

Generalized Terai's inequality for Stanley-Reisner ideals.

Abstract

The Leray number of an abstract simplicial complex is the minimal integer $d$ where its induced subcomplexes have trivial homology groups in dimension $d$ or greater. We give an upper bound on the Leray number of a complex in terms of how the facets are attached to each other. We also describe the structure of complexes for the equality of the bound that we found. Through the Stanley-Reisner correspondence, our results give an Eisenbud-Goto type inequality for any square-free monomial ideals. This generalizes Terai's result.