When is a squarefree monomial ideal of linear type?

Summary

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 to all squarefree monomial ideals by defining even walks in a simplicial complex. We show that simplicial complexes with no even walks have facet ideals that are of linear type, generalizing Villarreal’s work.

Rees algebras are of special interest in algebraic geometry and commutative algebra since they describe the blowing up of the spectrum of a ring along the subscheme defined by an ideal. The Rees algebra of an ideal can also be viewed as a quotient of a polynomial ring. If I is an ideal of a ring R, we denote the Rees algebra of I by R[I _t], and we can represent R[I _t] as S/J where S is a polynomial ring over R. The ideal J is called the defining ideal of R[I _t]. Finding generators of J is difficult and crucial for better understanding R[I _t]. Many authors have worked to gain better insight into these generators in special classes of ideals, such as those with special height, special embedding dimension and so on.