Scarf complexes of graphs and their powers
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908874040147968 |
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| author | Faridi, Sara Hà, Tài Huy Hibi, Takayuki Morey, Susan |
| author_facet | Faridi, Sara Hà, Tài Huy Hibi, Takayuki Morey, Susan |
| contents | Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scarf complexes of graphs and their powers Faridi, Sara Hà, Tài Huy Hibi, Takayuki Morey, Susan Commutative Algebra Combinatorics Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs. |
| title | Scarf complexes of graphs and their powers |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2403.05439 |