Monomial resolutions supported by simplicial trees

Abstract

We explore resolutions of monomial ideals supported by simplicial trees. We argue that, since simplicial trees are acyclic, the criterion of Bayer, Peeva and Sturmfels for checking whether a simplicial complex supports a free resolution of a monomial ideal reduces to checking that certain induced subcomplexes are connected. We then use results of Peeva and Velasco to show that every simplicial tree appears as the Scarf complex of a monomial ideal and hence supports a minimal resolution. We also provide a way to construct smaller Scarf ideals than those constructed by Peeva and Velasco.