Cohen-Macaulay squares of edge ideals
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914380150472704 |
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| author | Faridi, Sara Hibi, Takayuki |
| author_facet | Faridi, Sara Hibi, Takayuki |
| contents | Let $G$ be a finite graph and $I(G)$ its edge ideal. We give a full description of the Stanley--Reisner complex of the polarization of $I(G)^2$, naturally introducing the tools of Stanley--Reisner theory in the study of the algebraic behaviour of powers of edge ideals. As an application, we demonstrate how Reisner's criterion can be applied directly to check if $I(G)^2$ is Cohen--Macaulay. We can show that if $G$ belongs to the class of finite graphs which consists of cycles, whisker graphs, trees, connected chordal graphs and connected Cohen--Macaulay bipartite graphs, then the square $I(G)^2$ is Cohen--Macaulay if and only if either $G$ is the pentagon, the cycle of length $5$, or $G$ consists of exactly one edge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02605 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohen-Macaulay squares of edge ideals Faridi, Sara Hibi, Takayuki Commutative Algebra 06A11, 13D02 Let $G$ be a finite graph and $I(G)$ its edge ideal. We give a full description of the Stanley--Reisner complex of the polarization of $I(G)^2$, naturally introducing the tools of Stanley--Reisner theory in the study of the algebraic behaviour of powers of edge ideals. As an application, we demonstrate how Reisner's criterion can be applied directly to check if $I(G)^2$ is Cohen--Macaulay. We can show that if $G$ belongs to the class of finite graphs which consists of cycles, whisker graphs, trees, connected chordal graphs and connected Cohen--Macaulay bipartite graphs, then the square $I(G)^2$ is Cohen--Macaulay if and only if either $G$ is the pentagon, the cycle of length $5$, or $G$ consists of exactly one edge. |
| title | Cohen-Macaulay squares of edge ideals |
| topic | Commutative Algebra 06A11, 13D02 |
| url | https://arxiv.org/abs/2505.02605 |