DG-Sensitive Pruning & a Complete Classification of DG Trees and Cycles
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911649561051136 |
|---|---|
| author | Geller, Hugh Martin, Desiree Potts-Rubin, Henry |
| author_facet | Geller, Hugh Martin, Desiree Potts-Rubin, Henry |
| contents | 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 $\mathbb{F}$. In the language of combinatorics, this says 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 length of the longest path in $G$, where $I(G)$ is the edge ideal of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00591 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | DG-Sensitive Pruning & a Complete Classification of DG Trees and Cycles Geller, Hugh Martin, Desiree Potts-Rubin, Henry Commutative Algebra Combinatorics 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 $\mathbb{F}$. In the language of combinatorics, this says 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 length of the longest path in $G$, where $I(G)$ is the edge ideal of $G$. |
| title | DG-Sensitive Pruning & a Complete Classification of DG Trees and Cycles |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2502.00591 |