Edge ideals with linear quotients and without homological linear quotients
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| Format: | Preprint |
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2025
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| _version_ | 1866915677013540864 |
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| author | Chau, Trung Das, Kanoy Kumar Maithani, Aryaman |
| author_facet | Chau, Trung Das, Kanoy Kumar Maithani, Aryaman |
| contents | A monomial ideal $I$ is said to have homological linear quotients if for each $k\geq 0$, the homological shift ideal $\mathrm{HS}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear quotients, then $G$ is co-chordal. We construct a family of co-chordal graphs $\{\mathrm{H}_n^c\}_{n\geq 6}$ and propose a conjecture that an edge ideal $I(G)$ has homological linear quotients if and only if $G$ is co-chordal and $\mathrm{H}_n^c$-free for any $n\geq 6$. In this paper, we prove one direction of the conjecture. Moreover, we study possible patterns of pairs $(G,k)$ of a co-chordal graph $G$ and integer $k$ such that $\mathrm{HS}_k(I(G))$ has linear quotients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_11424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Edge ideals with linear quotients and without homological linear quotients Chau, Trung Das, Kanoy Kumar Maithani, Aryaman Commutative Algebra Combinatorics 13D02, 13F55, 05E40, 05C75 A monomial ideal $I$ is said to have homological linear quotients if for each $k\geq 0$, the homological shift ideal $\mathrm{HS}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear quotients, then $G$ is co-chordal. We construct a family of co-chordal graphs $\{\mathrm{H}_n^c\}_{n\geq 6}$ and propose a conjecture that an edge ideal $I(G)$ has homological linear quotients if and only if $G$ is co-chordal and $\mathrm{H}_n^c$-free for any $n\geq 6$. In this paper, we prove one direction of the conjecture. Moreover, we study possible patterns of pairs $(G,k)$ of a co-chordal graph $G$ and integer $k$ such that $\mathrm{HS}_k(I(G))$ has linear quotients. |
| title | Edge ideals with linear quotients and without homological linear quotients |
| topic | Commutative Algebra Combinatorics 13D02, 13F55, 05E40, 05C75 |
| url | https://arxiv.org/abs/2503.11424 |