Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers of Square-Free Monomial Ideals

TL;DR

This paper investigates the structure of square-free monomial ideals using well ordered covers and simplicial bouquets, exploring their Betti numbers and subadditivity properties through lattice complements and combinatorial methods.

Contribution

It introduces strongly disjoint simplicial bouquets and analyzes their role in the subadditivity of Betti numbers for square-free monomial ideals.

Findings

Well ordered covers guarantee nonvanishing Betti numbers in certain degrees.

Lattice complementation can decompose well ordered covers into covers of subideals.

Strongly disjoint simplicial bouquets can be identified in complexes and relate to subadditivity properties.

Abstract

Well ordered covers of square-free monomial ideals are subsets of the minimal generating set ordered in a certain way that give rise to a Lyubeznik resolution for the ideal, and have guaranteed nonvanishing Betti numbers in certain degrees. This paper is about square-free monomial ideals which have a well ordered cover. We consider the question of subadditivity of syzygies of square-free monomial ideals via complements in the lcm lattice of the ideal, and examine how lattice complementation breaks well ordered covers of the ideal into (well ordered) covers of subideals. We also introduce a family of well ordered covers called strongly disjoint sets of simplicial bouquets (generalizing work of Kimura on graphs), which are relatively easy to identify in simplicial complexes. We examine the subadditivity property via numerical characteristics of these bouquets.