DG-Sensitive Pruning & a Complete Classification of DG Trees and Cycles
Hugh Geller, Desiree Martin, Henry Potts-Rubin
TL;DR
The paper investigates when minimal free resolutions of squarefree monomial ideals admit a differential graded (dg) algebra structure and proves that dg-ness is preserved under Boocher-type pruning. By combining Lyubeznik resolutions, discrete Morse theory, and the pruning framework, it provides complete classifications for when edge ideals of graphs (specifically trees and cycles) yield dg algebra resolutions, in terms of graph diameter. It constructs an explicit dg algebra resolution for diameter-4 trees via a mapping-cone approach and uses pruning to derive obstructions that rule out dg-structures for larger trees and cycles, including all cycles with at least seven vertices. These results link combinatorial graph invariants to strong algebraic structures on resolutions, with facet-induced subcomplexes preserving dg-ness.
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 $\mathbb{F}$. As an application, we show that if $Q/\mathcal{F}(Δ)$, the quotient of the ambient polynomial ring by the facet ideal $\mathcal{F}(Δ)$ of a simplicial complex $Δ$, is minimally resolved by a dg algebra, then so is the quotient by the facet ideal of each facet-induced subcomplex of $Δ$ (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 diameter of $G$, where $I_G$ is the edge ideal of $G$.