Some criteria for Gorensteinness via Gorenstein projective cotorsion pairs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910037647032320 |
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| author | Dey, Souvik Liu, Jian Lu, Xue-Song |
| author_facet | Dey, Souvik Liu, Jian Lu, Xue-Song |
| contents | Let $R$ be a noetherian algebra over a Cohen--Macaulay ring admitting a canonical module, and assume that $R$ is maximal Cohen--Macaulay over the base ring. We provide a characterization of when $R$ is left weakly Gorenstein. We further show that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\Ext$-orthogonal class of the thick subcategory generated by finitely generated $R$-modules of finite projective or finite injective dimension. As a consequence, finitely generated Gorenstein projective $R$-modules generate a hereditary cotorsion pair. Moreover, we show that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01424 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some criteria for Gorensteinness via Gorenstein projective cotorsion pairs Dey, Souvik Liu, Jian Lu, Xue-Song Rings and Algebras Commutative Algebra Representation Theory Let $R$ be a noetherian algebra over a Cohen--Macaulay ring admitting a canonical module, and assume that $R$ is maximal Cohen--Macaulay over the base ring. We provide a characterization of when $R$ is left weakly Gorenstein. We further show that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\Ext$-orthogonal class of the thick subcategory generated by finitely generated $R$-modules of finite projective or finite injective dimension. As a consequence, finitely generated Gorenstein projective $R$-modules generate a hereditary cotorsion pair. Moreover, we show that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension. |
| title | Some criteria for Gorensteinness via Gorenstein projective cotorsion pairs |
| topic | Rings and Algebras Commutative Algebra Representation Theory |
| url | https://arxiv.org/abs/2603.01424 |