On squarefree powers of simplicial trees

TL;DR

This paper investigates the algebraic properties of squarefree powers of facet ideals from simplicial trees, focusing on their resolutions, regularity, and syzygies, with a combinatorial formula for specific path ideals.

Contribution

It provides new insights into the linearity, regularity, and syzygy generation of squarefree powers of facet ideals in simplicial trees and path graphs.

Findings

Conditions for linearity of minimal free resolutions

Explicit combinatorial formula for regularity of t-path ideals

Characterization of generators of the first syzygy module

Abstract

In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first syzygy module of squarefree powers of a simplicial tree is generated by linear relations. Finally, we provide a combinatorial formula for the regularity of the squarefree powers of $t$-path ideals of path graphs.