Projective dimension of weakly chordal graphic arrangements
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909617153376256 |
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| author | Abe, Takuro Kühne, Lukas Mücksch, Paul Mühlherr, Leonie |
| author_facet | Abe, Takuro Kühne, Lukas Mücksch, Paul Mühlherr, Leonie |
| contents | A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_06021 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Projective dimension of weakly chordal graphic arrangements Abe, Takuro Kühne, Lukas Mücksch, Paul Mühlherr, Leonie Combinatorics 52C35, 32S22, 20F55, 51F15 A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices. |
| title | Projective dimension of weakly chordal graphic arrangements |
| topic | Combinatorics 52C35, 32S22, 20F55, 51F15 |
| url | https://arxiv.org/abs/2307.06021 |