Cohen-Macaulay Properties of Square-Free Monomial Ideals

TL;DR

This paper explores the algebraic properties of simplicial complexes through their facet ideals, introducing new classes of Cohen-Macaulay ideals and criteria for their Cohen-Macaulayness, generalizing graph theory concepts.

Contribution

It introduces a broad class of Cohen-Macaulay square-free monomial ideals and provides a criterion for Cohen-Macaulayness of facet ideals of simplicial trees.

Findings

Defined a large class of Cohen-Macaulay square-free monomial ideals

Established a criterion for Cohen-Macaulayness of facet ideals of simplicial trees

Generalized graph theory concepts to simplicial complexes

Abstract

In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.