Homological dimensions, the Gorenstein property, and special cases of some conjectures
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909185946419200 |
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| author | Dey, Souvik Holanda, Rafael Miranda-Neto, Cleto B. |
| author_facet | Dey, Souvik Holanda, Rafael Miranda-Neto, Cleto B. |
| contents | Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we investigate the impact of the finiteness of certain homological dimensions of $M$ if $R$ is local, mainly when $R$ is Cohen-Macaulay and with a partial focus on duals. Along the way, we produce various freeness criteria for modules. Finally, we give applications, including characterizations of when $R$ is Gorenstein (and other ring-theoretic properties as well, sometimes in the prime characteristic setting), particularly by means of its anticanonical module, and in addition we address special cases of some long-standing conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on normal modules in case the module of differentials is almost Cohen-Macaulay. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00152 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homological dimensions, the Gorenstein property, and special cases of some conjectures Dey, Souvik Holanda, Rafael Miranda-Neto, Cleto B. Commutative Algebra Primary 13D05, 13C10, 13H10, 13N15, Secondary 13D02, 13D07, 13C14 Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we investigate the impact of the finiteness of certain homological dimensions of $M$ if $R$ is local, mainly when $R$ is Cohen-Macaulay and with a partial focus on duals. Along the way, we produce various freeness criteria for modules. Finally, we give applications, including characterizations of when $R$ is Gorenstein (and other ring-theoretic properties as well, sometimes in the prime characteristic setting), particularly by means of its anticanonical module, and in addition we address special cases of some long-standing conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on normal modules in case the module of differentials is almost Cohen-Macaulay. |
| title | Homological dimensions, the Gorenstein property, and special cases of some conjectures |
| topic | Commutative Algebra Primary 13D05, 13C10, 13H10, 13N15, Secondary 13D02, 13D07, 13C14 |
| url | https://arxiv.org/abs/2405.00152 |