On modules whose dual is of finite Gorenstein dimension
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918188478889984 |
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| author | Mendoza-Rubio, Victor D. Jorge-Pérez, Victor H. |
| author_facet | Mendoza-Rubio, Victor D. Jorge-Pérez, Victor H. |
| contents | In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the freeness or totally reflexivity of modules. For instance, we provide a generalization of a celebrated theorem by Auslander and Bridger, obtain criteria for the totally reflexivity of modules over Cohen-Macaulay rings as well as of locally totally reflexive modules on the punctured spectrum, and recover a result by Araya. Moreover, we prove that the Auslander-Reiten conjecture holds true for all finitely generated modules $M$ over a commutative Noetherian ring $R$ such that $\operatorname{G-dim}_R(\operatorname{Hom}_R(M,R))<\infty$ and $\operatorname{pd}_R(\operatorname{Hom}_R(M,M))<\infty$. Additionally, we derive Gorenstein criteria under the condition that the dual of certain modules is of finite Gorenstein dimension. Furthermore, we explore some applications in the theory of the modules of Kähler differentials of order $n\geq 1$, specifically concerning the $k$-torsionfreeness of these modules and the Herzog-Vasconcelos conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06124 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On modules whose dual is of finite Gorenstein dimension Mendoza-Rubio, Victor D. Jorge-Pérez, Victor H. Commutative Algebra Primary: 13D45, 13D07, 13C10, 13C14, Secondary: 13D05, 13D02, 13H10, 14B15, 14B05 In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the freeness or totally reflexivity of modules. For instance, we provide a generalization of a celebrated theorem by Auslander and Bridger, obtain criteria for the totally reflexivity of modules over Cohen-Macaulay rings as well as of locally totally reflexive modules on the punctured spectrum, and recover a result by Araya. Moreover, we prove that the Auslander-Reiten conjecture holds true for all finitely generated modules $M$ over a commutative Noetherian ring $R$ such that $\operatorname{G-dim}_R(\operatorname{Hom}_R(M,R))<\infty$ and $\operatorname{pd}_R(\operatorname{Hom}_R(M,M))<\infty$. Additionally, we derive Gorenstein criteria under the condition that the dual of certain modules is of finite Gorenstein dimension. Furthermore, we explore some applications in the theory of the modules of Kähler differentials of order $n\geq 1$, specifically concerning the $k$-torsionfreeness of these modules and the Herzog-Vasconcelos conjecture. |
| title | On modules whose dual is of finite Gorenstein dimension |
| topic | Commutative Algebra Primary: 13D45, 13D07, 13C10, 13C14, Secondary: 13D05, 13D02, 13H10, 14B15, 14B05 |
| url | https://arxiv.org/abs/2312.06124 |