Non-existence of Ulrich modules over Cohen-Macaulay local rings
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912270348451840 |
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| author | Iyengar, Srikanth B. Ma, Linquan Walker, Mark E. Zhuang, Ziquan |
| author_facet | Iyengar, Srikanth B. Ma, Linquan Walker, Mark E. Zhuang, Ziquan |
| contents | Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-existence of Ulrich modules over Cohen-Macaulay local rings Iyengar, Srikanth B. Ma, Linquan Walker, Mark E. Zhuang, Ziquan Commutative Algebra Algebraic Geometry 13C13 (primary), 13H10, 13C14, 14F06 (secondary) Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal. |
| title | Non-existence of Ulrich modules over Cohen-Macaulay local rings |
| topic | Commutative Algebra Algebraic Geometry 13C13 (primary), 13H10, 13C14, 14F06 (secondary) |
| url | https://arxiv.org/abs/2403.15566 |