Abstract 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 () Q / F (Δ), the quotient of the ambient polynomial ring by the facet ideal F () 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 length of the longest path in G, where I (G) is the edge ideal of G.
Geller et al. (Fri,) studied this question.