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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.16643 |
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| _version_ | 1866914780348940288 |
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| author | Adiprasito, Karim Björner, Anders Hakavuori, Joel Margaritis, Minas Welker, Volkmar |
| author_facet | Adiprasito, Karim Björner, Anders Hakavuori, Joel Margaritis, Minas Welker, Volkmar |
| contents | We show that the maximal shifts in the minimal free resolution of the quotients of a polynomial ring by a monomial ideal are subadditive as a function of the homological degree. This answers a question that has received some attention in recent years. To do so, we define and study a new model for the homology of posets, given by the so called synor complex. We also introduce an Eilenberg-Zilber type shuffle product on the simplicial chain complex of lattices.
Combining these concepts we prove that the existence of a non-zero homology class for a lattice forces certain non-zero homology classes in lower intervals. This result then translates into properties of the minimal free resolution. In particular, it implies a generalization of the original question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16643 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subadditivity of shifts, Eilenberg-Zilber shuffle products and homology of lattices Adiprasito, Karim Björner, Anders Hakavuori, Joel Margaritis, Minas Welker, Volkmar Commutative Algebra Combinatorics We show that the maximal shifts in the minimal free resolution of the quotients of a polynomial ring by a monomial ideal are subadditive as a function of the homological degree. This answers a question that has received some attention in recent years. To do so, we define and study a new model for the homology of posets, given by the so called synor complex. We also introduce an Eilenberg-Zilber type shuffle product on the simplicial chain complex of lattices. Combining these concepts we prove that the existence of a non-zero homology class for a lattice forces certain non-zero homology classes in lower intervals. This result then translates into properties of the minimal free resolution. In particular, it implies a generalization of the original question. |
| title | Subadditivity of shifts, Eilenberg-Zilber shuffle products and homology of lattices |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2404.16643 |