On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917195439669248 |
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| author | Nguyen, Van C. Veliche, Oana |
| author_facet | Nguyen, Van C. Veliche, Oana |
| contents | Let $R$ be any noetherian local ring with residue field $k$, and $A$ the homology of the Koszul complex on a minimal set of generators of the maximal ideal of $R$. In this paper, we show that a minimal free resolution of $k$ over $R$ can be obtained from a graded minimal free resolution of $k$ over $A$. More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of $k$ over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of $k$ over a noetherian local ring of codepth 3 of class $T$ in terms of Koszul blocks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$ Nguyen, Van C. Veliche, Oana Commutative Algebra Rings and Algebras 13D02, 13D07, 13H10, 18G10, 18G35 Let $R$ be any noetherian local ring with residue field $k$, and $A$ the homology of the Koszul complex on a minimal set of generators of the maximal ideal of $R$. In this paper, we show that a minimal free resolution of $k$ over $R$ can be obtained from a graded minimal free resolution of $k$ over $A$. More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of $k$ over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of $k$ over a noetherian local ring of codepth 3 of class $T$ in terms of Koszul blocks. |
| title | On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$ |
| topic | Commutative Algebra Rings and Algebras 13D02, 13D07, 13H10, 18G10, 18G35 |
| url | https://arxiv.org/abs/2506.21505 |