Notes on modules of finite injective dimension
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913160478326784 |
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| author | Asgharzadeh, Mohsen |
| author_facet | Asgharzadeh, Mohsen |
| contents | Motivated by the Bass conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond the Cohen--Macaulay property, the existence of such modules forces further conditions on the ring, including reducedness, normality, being an integral domain, and various conditions such as being complete intersection or Gorenstein.
We also address the reflexivity and torsionlessness of modules of finite injective dimension, showing that these properties force the ring to be quasi-normal. In the same vein, we investigate the injective dimension of tensor products and endomorphism rings. Finally, we study the behavior of R when a high syzygy of the residue field surjects onto a nonzero module of finite injective dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01105 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Notes on modules of finite injective dimension Asgharzadeh, Mohsen Commutative Algebra Motivated by the Bass conjecture, we study finitely generated modules of finite injective dimension and the additional constraints they impose on the ambient ring. Beyond the Cohen--Macaulay property, the existence of such modules forces further conditions on the ring, including reducedness, normality, being an integral domain, and various conditions such as being complete intersection or Gorenstein. We also address the reflexivity and torsionlessness of modules of finite injective dimension, showing that these properties force the ring to be quasi-normal. In the same vein, we investigate the injective dimension of tensor products and endomorphism rings. Finally, we study the behavior of R when a high syzygy of the residue field surjects onto a nonzero module of finite injective dimension. |
| title | Notes on modules of finite injective dimension |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2301.01105 |