On direct summands of syzygies of the residue field of a local ring
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912673918091264 |
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| author | Cuong, Doan Trung Kobayashi, Toshinori |
| author_facet | Cuong, Doan Trung Kobayashi, Toshinori |
| contents | We investigate local rings in which a syzygy of the residue field occurs as a direct summand of another syzygy of the field. This class of local rings includes Golod rings, Burch rings and non-trivial fiber products of local rings. For such rings, we prove that the Betti sequence of any finitely generated module is eventually periodically non-decreasing. As an application, we confirm the Tachikawa conjecture for all Cohen-Macaulay local rings satisfying this syzygy condition. In the second part of the paper, we show that a recursive direct sum decomposition of the syzygy of the residue field characterizes Golod rings, thereby establishing the converse to a recent theorem of Cuong-Dao-Eisenbud-Kobayashi-Polini-Ulrich [10]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On direct summands of syzygies of the residue field of a local ring Cuong, Doan Trung Kobayashi, Toshinori Commutative Algebra Primary: 13D02. Secondary: 13D07, 13H10 We investigate local rings in which a syzygy of the residue field occurs as a direct summand of another syzygy of the field. This class of local rings includes Golod rings, Burch rings and non-trivial fiber products of local rings. For such rings, we prove that the Betti sequence of any finitely generated module is eventually periodically non-decreasing. As an application, we confirm the Tachikawa conjecture for all Cohen-Macaulay local rings satisfying this syzygy condition. In the second part of the paper, we show that a recursive direct sum decomposition of the syzygy of the residue field characterizes Golod rings, thereby establishing the converse to a recent theorem of Cuong-Dao-Eisenbud-Kobayashi-Polini-Ulrich [10]. |
| title | On direct summands of syzygies of the residue field of a local ring |
| topic | Commutative Algebra Primary: 13D02. Secondary: 13D07, 13H10 |
| url | https://arxiv.org/abs/2510.24220 |