DG-Sensitive Pruning & a Complete Classification of DG Trees and Cycles

TL;DR

This paper proves that if a minimal free resolution of a squarefree monomial ideal admits a dg algebra structure, then so do all its prunings, leading to a complete classification of certain graphs based on their resolutions.

Contribution

It establishes that dg algebra structures are preserved under pruning of resolutions and classifies trees and cycles with dg algebra-resolved edge ideals.

Findings

Pruning preserves dg algebra structures in resolutions.

Complete classification of trees and cycles based on longest path length.

Resolution properties depend on the structure of the underlying graph.

Abstract

Given a squarefree monomial ideal $I$ of a polynomial ring $Q$, we show that if the minimal free resolution $\mathbb{F}$ of $Q/I$ admits the structure of a differential graded (dg) algebra, then so does any ``pruning" of $\mathbb{F}$. In the language of combinatorics, this says that if $Q/\mathcal{F}(\Delta)$, the quotient of the ambient polynomial ring by the facet ideal $\mathcal{F}(\Delta)$ of a simplicial complex $\Delta$, is minimally resolved by a dg algebra, then so is the quotient by the facet ideal of each facet-induced subcomplex of $\Delta$ (over the smaller polynomial ring). Along with techniques from discrete Morse theory and homological algebra, this allows us to give complete classifications of the trees and cycles $G$ with $Q/I(G)$ minimally resolved by a dg algebra in terms of the length of the longest path in $G$, where $I(G)$ is the edge ideal of $G$.