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

Abstract Given a squarefree monomial ideal I of a polynomial ring Q , we show that if the minimal free resolution $$\mathbb {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> of Q / I admits the structure of a differential graded (dg) algebra, then so does any “pruning” of $$\mathbb {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . In the language of combinatorics, this says that if $$Q/\mathcal {F}(\Delta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the quotient of the ambient polynomial ring by the facet ideal $$\mathcal {F}(\Delta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of a simplicial complex $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> , is minimally resolved by a dg algebra, then so is the quotient by the facet ideal of each facet-induced subcomplex of $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> (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 .