Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911824830529536 |
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| author | Nguyen, Van C. Veliche, Oana |
| author_facet | Nguyen, Van C. Veliche, Oana |
| contents | Let $(R,\mathfrak m, \mathsf k)$ be a complete intersection local ring, $K$ be the Koszul complex on a minimal set of generators of $\mathfrak m$, and $A=H(K)$ be its homology algebra. We establish exact sequences involving direct sums of the components of $A$ and express the images of the maps of these sequences as homologies of iterated mapping cones built on $K$. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field $\mathsf k$ over $R$, independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_02606 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings Nguyen, Van C. Veliche, Oana Commutative Algebra K-Theory and Homology Rings and Algebras 13D02, 13D07, 13H10, 18G10, 18G35 Let $(R,\mathfrak m, \mathsf k)$ be a complete intersection local ring, $K$ be the Koszul complex on a minimal set of generators of $\mathfrak m$, and $A=H(K)$ be its homology algebra. We establish exact sequences involving direct sums of the components of $A$ and express the images of the maps of these sequences as homologies of iterated mapping cones built on $K$. As an application of this iterated mapping cone construction, we recover a minimal free resolution of the residue field $\mathsf k$ over $R$, independent from the well-known resolution constructed by Tate by adjoining variables and killing cycles. Through our construction, the differential maps can be expressed explicitly as blocks of matrices, arranged in some combinatorial patterns. |
| title | Iterated Mapping Cones on the Koszul Complex and Their Application to Complete Intersection Rings |
| topic | Commutative Algebra K-Theory and Homology Rings and Algebras 13D02, 13D07, 13H10, 18G10, 18G35 |
| url | https://arxiv.org/abs/2212.02606 |