When is a Squarefree Monomial Ideal of Linear Type ?

TL;DR

This paper generalizes Villarreal's combinatorial characterization of the equations of Rees algebras from quadratic squarefree monomial ideals to all squarefree monomial ideals using simplicial complexes.

Contribution

It extends the concept of even walks from graphs to simplicial complexes, broadening the understanding of the algebraic structure of squarefree monomial ideals.

Findings

Generalized Villarreal's description to all squarefree monomial ideals.

Introduced a new definition of even walks in simplicial complexes.

Provided a combinatorial framework for analyzing Rees algebras of these ideals.

Abstract

In 1995 Villarreal gave a combinatorial description of the equations of Rees algebras of quadratic squarefree monomial ideals. His description was based on the concept of closed even walks in a graph. In this paper we will generalize his results for all squarefree monomial ideals by using a definition of even walks in a simplicial complex.