Regular Edges, Matchings and Hilbert Series
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913610782998528 |
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| author | Brennan, Joseph Morey, Susan |
| author_facet | Brennan, Joseph Morey, Susan |
| contents | When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, particularly Property $P$ on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on $R/I(G)$. Under a mild separability assumption, we identify when such elements can be combined to form a regular sequence. Using these regular sequences, we show that the Hilbert series and corresponding $h$-vector can be calculated from a related graph using a simplified calculation on the $f$-vector, or independence vector, of the related graph. In the case when the graph is Cohen-Macaulay with a perfect matching of regular edges satisfying the separability criterion, the $h$-vector of $R/I(G)$ will be precisely the $f$-vector of the Stanley-Reisner complex of a graph with half as many vertices as $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10335 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular Edges, Matchings and Hilbert Series Brennan, Joseph Morey, Susan Commutative Algebra 13F55, 13D40, 05E40 When $I$ is the edge ideal of a graph $G$, we use combinatorial properities, particularly Property $P$ on connectivity of neighbors of an edge, to classify when a binomial sum of vertices is a regular element on $R/I(G)$. Under a mild separability assumption, we identify when such elements can be combined to form a regular sequence. Using these regular sequences, we show that the Hilbert series and corresponding $h$-vector can be calculated from a related graph using a simplified calculation on the $f$-vector, or independence vector, of the related graph. In the case when the graph is Cohen-Macaulay with a perfect matching of regular edges satisfying the separability criterion, the $h$-vector of $R/I(G)$ will be precisely the $f$-vector of the Stanley-Reisner complex of a graph with half as many vertices as $G$. |
| title | Regular Edges, Matchings and Hilbert Series |
| topic | Commutative Algebra 13F55, 13D40, 05E40 |
| url | https://arxiv.org/abs/2412.10335 |