Shedding vertices and Ass-decomposable monomial ideals

TL;DR

This paper introduces ass-decomposable monomial ideals, generalizing vertex decomposable simplicial complexes, and provides formulas and bounds for their algebraic invariants such as depth and regularity.

Contribution

It defines ass-decomposable monomial ideals and establishes their recursive structure, enabling explicit calculations of depth and bounds for regularity.

Findings

Provided a formula for the depth of ass-decomposable ideals.

Established an upper bound for the regularity in the squarefree case.

Connected algebraic properties of ideals with combinatorial structures.

Abstract

The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner ideals of vertex decomposable simplicial complexes. The recursive structure of ass-decomposable monomial ideals allows us to find a simple formula for the depth, and in squarefree case, an upper bound for the regularity of such ideals.