Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910681180143616 |
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| author | Das, Kanoy Kumar Roy, Amit Saha, Kamalesh |
| author_facet | Das, Kanoy Kumar Roy, Amit Saha, Kamalesh |
| contents | Let $I(G)^{[k]}$ denote the $k^{th}$ square-free power of the edge ideal $I(G)$ of a graph $G$. In this article, we provide a precise formula for the depth of $I(G)^{[k]}$ when $G$ is a Cohen-Macaulay forest. Using this, we show that for a Cohen-Macaulay forest $G$, the $k^{th}$ square-free power of $I(G)$ is always Cohen-Macaulay, which is quite surprising since all ordinary powers of $I(G)$ can never be Cohen-Macaulay unless $G$ is a disjoint union of edges. Next, we give an exact formula for the regularity and tight bounds on the depth of square-free powers of edge ideals of cycles. In the case of whiskered cycles, we obtain tight bounds on the regularity and depth of square-free powers, which aids in identifying when such ideals have linear resolutions. Additionally, we compute depth of $I(G)^{[2]}$ when $G$ is a cycle or whiskered cycle, and regularity of $I(G)^{[2]}$ when $G$ is a whiskered cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles Das, Kanoy Kumar Roy, Amit Saha, Kamalesh Commutative Algebra Combinatorics Primary: 13C15, 05E40, 13D02, Secondary: 13H10, 05C70 Let $I(G)^{[k]}$ denote the $k^{th}$ square-free power of the edge ideal $I(G)$ of a graph $G$. In this article, we provide a precise formula for the depth of $I(G)^{[k]}$ when $G$ is a Cohen-Macaulay forest. Using this, we show that for a Cohen-Macaulay forest $G$, the $k^{th}$ square-free power of $I(G)$ is always Cohen-Macaulay, which is quite surprising since all ordinary powers of $I(G)$ can never be Cohen-Macaulay unless $G$ is a disjoint union of edges. Next, we give an exact formula for the regularity and tight bounds on the depth of square-free powers of edge ideals of cycles. In the case of whiskered cycles, we obtain tight bounds on the regularity and depth of square-free powers, which aids in identifying when such ideals have linear resolutions. Additionally, we compute depth of $I(G)^{[2]}$ when $G$ is a cycle or whiskered cycle, and regularity of $I(G)^{[2]}$ when $G$ is a whiskered cycle. |
| title | Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles |
| topic | Commutative Algebra Combinatorics Primary: 13C15, 05E40, 13D02, Secondary: 13H10, 05C70 |
| url | https://arxiv.org/abs/2409.06021 |