Minimal resolutions of lattice ideals
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910642185699328 |
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| author | Li, Yupeng Miller, Ezra Ordog, Erika |
| author_facet | Li, Yupeng Miller, Ezra Ordog, Erika |
| contents | A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial description as a sum over lattice paths in $\mathbb{Z}^n$ of weights that come from sequences of faces in simplicial complexes indexed by lattice points. Over a field of any characteristic, a non-canonical but simpler resolution is constructed by selecting choices of higher-dimensional analogues of spanning trees along lattice paths. These constructions generalize sylvan resolutions for monomial ideals by lifting them equivariantly to lattice modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_09557 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Minimal resolutions of lattice ideals Li, Yupeng Miller, Ezra Ordog, Erika Commutative Algebra Combinatorics 13F65, 13D02, 05E40, 13F20, 13C13, 20M25 (Primary), 05E45, 68W30, 20M14 (Secondary) A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial description as a sum over lattice paths in $\mathbb{Z}^n$ of weights that come from sequences of faces in simplicial complexes indexed by lattice points. Over a field of any characteristic, a non-canonical but simpler resolution is constructed by selecting choices of higher-dimensional analogues of spanning trees along lattice paths. These constructions generalize sylvan resolutions for monomial ideals by lifting them equivariantly to lattice modules. |
| title | Minimal resolutions of lattice ideals |
| topic | Commutative Algebra Combinatorics 13F65, 13D02, 05E40, 13F20, 13C13, 20M25 (Primary), 05E45, 68W30, 20M14 (Secondary) |
| url | https://arxiv.org/abs/2208.09557 |