On modules with finite reducing Gorenstein dimension
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866910837570011136 |
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| author | Araya, Tokuji Celikbas, Olgur Cook, Jesse Kobayashi, Toshinori |
| author_facet | Araya, Tokuji Celikbas, Olgur Cook, Jesse Kobayashi, Toshinori |
| contents | If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective dimension, and hence a result of Foxby implies that $R$ is Gorenstein. We investigate whether the same conclusion holds for nonzero finitely generated modules that have finite injective dimension and finite reducing Gorenstein dimension, where the reducing Gorenstein dimension is a finer invariant than the classical Gorenstein dimension, in general. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_00253 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On modules with finite reducing Gorenstein dimension Araya, Tokuji Celikbas, Olgur Cook, Jesse Kobayashi, Toshinori Commutative Algebra If $M$ is a nonzero finitely generated module over a commutative Noetherian local ring $R$ such that $M$ has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that $M$ has finite projective dimension, and hence a result of Foxby implies that $R$ is Gorenstein. We investigate whether the same conclusion holds for nonzero finitely generated modules that have finite injective dimension and finite reducing Gorenstein dimension, where the reducing Gorenstein dimension is a finer invariant than the classical Gorenstein dimension, in general. |
| title | On modules with finite reducing Gorenstein dimension |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2103.00253 |