Given a squarefree monomial ideal I of a polynomial ring Q, we show that if the minimal free resolution <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation> of Q/I admits the structure of a differential graded (dg) algebra, then so does any “pruning” of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">F</mi> </math></EquationSource> </InlineEquation>. In the language of combinatorics, this says that if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Q/\mathcal {F}(\Delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mo stretchy="false">/</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the quotient of the ambient polynomial ring by the facet ideal <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}(\Delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of a simplicial complex <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation>, is minimally resolved by a dg algebra, then so is the quotient by the facet ideal of each facet-induced subcomplex of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> (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.

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

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2026年5月5日

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Hugh Geller,
Desiree Martin,
Henry Potts-Rubin

摘要

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