Koszul complexes and relative homological algebra of functors over posets
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913325430865920 |
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| author | Chacholski, Wojciech Guidolin, Andrea Ren, Isaac Scolamiero, Martina Tombari, Francesca |
| author_facet | Chacholski, Wojciech Guidolin, Andrea Ren, Isaac Scolamiero, Martina Tombari, Francesca |
| contents | Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_05923 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Koszul complexes and relative homological algebra of functors over posets Chacholski, Wojciech Guidolin, Andrea Ren, Isaac Scolamiero, Martina Tombari, Francesca Algebraic Topology 18G25, 55N31 Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors. |
| title | Koszul complexes and relative homological algebra of functors over posets |
| topic | Algebraic Topology 18G25, 55N31 |
| url | https://arxiv.org/abs/2209.05923 |