On squarefree powers of simplicial trees

TL;DR

This paper investigates squarefree powers $I(\Delta)^{[k]}$ of facet ideals of simplicial trees, linking them to $k$-matchings and studying when their minimal free resolutions are linear and what their regularities are. It shows that if $I(\Delta)$ has a linear resolution, then the intersection property forces $\nu(\Delta)\le 2$, so only $I(\Delta)^{[1]}$ and $I(\Delta)^{[2]}$ can occur, with $I(\Delta)^{[2]}$ retaining linearity. It further analyzes higher squarefree powers, giving a conditional linear-quotient result for broom graphs and path graphs, and proves a sharp combinatorial formula for the regularity of $t$-path ideals on path graphs: $\operatorname{reg}(R/I_{n,t}^{[k+1]}) = kt+(t-1)\nu_1(\Gamma_{n-kt,t})$. The work culminates in several open questions and conjectures about extending these bounds and descriptions to broader classes of simplicial trees and their squarefree powers.

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.

Figures (2)

• Figure 2: A rooted tree.
• Figure 3: A broom graph.

Theorems & Definitions (44)

• Example 1.1
• Proposition 1.2
• proof
• Corollary 1.3
• Proposition 1.4
• proof
• Proposition 2.1
• proof
• Theorem 2.2
• proof