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| Autors principals: | , |
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| Format: | Preprint |
| Publicat: |
2022
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| Matèries: | |
| Accés en línia: | https://arxiv.org/abs/2212.05708 |
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| _version_ | 1866910596573691904 |
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| author | Saha, Kamalesh Sengupta, Indranath |
| author_facet | Saha, Kamalesh Sengupta, Indranath |
| contents | Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. We prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only "biconnected graphs with some whisker attached" and this done by investigating the initial ideals. We give several necessary conditions for a binomial edge ideal to be Cohen-Macaulay in terms of smaller graphs. Also, under a hypothesis, we give a sufficient condition for Cohen-Macaulayness of binomial edge ideals in terms of blocks of graphs. Moreover, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than $5$ or equal to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_05708 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cohen-Macaulay Property of Binomial Edge Ideals with Girth of Graphs Saha, Kamalesh Sengupta, Indranath Commutative Algebra 13C14, 13F65, 13F55, 05E40, 05C25 Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. We prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only "biconnected graphs with some whisker attached" and this done by investigating the initial ideals. We give several necessary conditions for a binomial edge ideal to be Cohen-Macaulay in terms of smaller graphs. Also, under a hypothesis, we give a sufficient condition for Cohen-Macaulayness of binomial edge ideals in terms of blocks of graphs. Moreover, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than $5$ or equal to infinity. |
| title | Cohen-Macaulay Property of Binomial Edge Ideals with Girth of Graphs |
| topic | Commutative Algebra 13C14, 13F65, 13F55, 05E40, 05C25 |
| url | https://arxiv.org/abs/2212.05708 |