Monomial Resolutions Supported By Simplicial Trees

TL;DR

This paper investigates how simplicial trees can support monomial ideal resolutions, showing their connection to Scarf complexes and providing methods to construct minimal resolutions more efficiently.

Contribution

It demonstrates that simplicial trees support minimal resolutions of monomial ideals and introduces a new approach to construct smaller Scarf ideals.

Findings

Simplicial trees are acyclic and support resolutions if certain subcomplexes are connected.

Every simplicial tree can be realized as the Scarf complex of a monomial ideal.

New methods are provided to construct smaller Scarf ideals than previously known.

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 if 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.