DG-sensitive pruning & a complete classification of DG trees and cycles

Given a squarefree monomial ideal I of a polynomial ring Q , we show that if the minimal free resolution F F of Q / I admits the structure of a differential graded (dg) algebra, then so does any “pruning” of F F . In the language of combinatorics, this says that if Q/F(Δ ) Q / F ( Δ ) , the quotient of the ambient polynomial ring by the facet ideal F(Δ ) F ( Δ ) of a simplicial complex Δ Δ , is minimally resolved by a dg algebra, then so is the quotient by the facet ideal of each facet-induced subcomplex of Δ 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 length of the longest path in G , where I ( G ) is the edge ideal of G .