Gapfree graphs and powers of edge ideals with linear quotients
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arXiv
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| Format: | Preprint |
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2024
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| author | Erey, Nursel Faridi, Sara Hà, Tài Huy Hibi, Takayuki Kara, Selvi Morey, Susan |
| author_facet | Erey, Nursel Faridi, Sara Hà, Tài Huy Hibi, Takayuki Kara, Selvi Morey, Susan |
| contents | Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gapfree graphs and powers of edge ideals with linear quotients Erey, Nursel Faridi, Sara Hà, Tài Huy Hibi, Takayuki Kara, Selvi Morey, Susan Commutative Algebra Combinatorics 05E40, 13D02 Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$. |
| title | Gapfree graphs and powers of edge ideals with linear quotients |
| topic | Commutative Algebra Combinatorics 05E40, 13D02 |
| url | https://arxiv.org/abs/2412.06467 |