Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929266530189312 |
|---|---|
| author | Ghosh, Dipankar Puthenpurakal, Tony J. |
| author_facet | Ghosh, Dipankar Puthenpurakal, Tony J. |
| contents | The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let $R$ be a commutative Noetherian local ring of dimension $d$. In the 1st part, it is proved that $R$ is Gorenstein if and only if it admits a nonzero CM (Cohen-Macaulay) module $M$ of finite Gorenstein dimension $g$ such that ${\rm type}(M) \le μ( {\rm Ext}_R^g(M,R) )$ (e.g., ${\rm type}(M)=1$). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero $R$-module $M$ of depth $\ge d - 1$ such that the injective dimensions of $M$, ${\rm Hom}_R(M,M)$ and ${\rm Ext}_R^1(M,M)$ are finite, then $M$ has finite projective dimension and $R$ is Gorenstein. In the 2nd part, we assume that $R$ is CM with a canonical module $ω$. For CM $R$-modules $M$ and $N$, we show that the vanishing of one of the following implies the same for others: ${\rm Ext}_R^{\gg 0}(M,N^{+})$, ${\rm Ext}_R^{\gg 0}(N,M^{+})$ and ${\rm Tor}_{\gg 0}^R(M,N)$, where $M^{+}$ denotes ${\rm Ext}_R^{d-\dim(M)}(M,ω)$. This strengthens a result of Huneke and Jorgensen. Furthermore, we prove a similar result for Tate cohomologies under the additional condition that $R$ is Gorenstein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06267 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology Ghosh, Dipankar Puthenpurakal, Tony J. Commutative Algebra 13D05, 13D07, 13H10 The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let $R$ be a commutative Noetherian local ring of dimension $d$. In the 1st part, it is proved that $R$ is Gorenstein if and only if it admits a nonzero CM (Cohen-Macaulay) module $M$ of finite Gorenstein dimension $g$ such that ${\rm type}(M) \le μ( {\rm Ext}_R^g(M,R) )$ (e.g., ${\rm type}(M)=1$). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero $R$-module $M$ of depth $\ge d - 1$ such that the injective dimensions of $M$, ${\rm Hom}_R(M,M)$ and ${\rm Ext}_R^1(M,M)$ are finite, then $M$ has finite projective dimension and $R$ is Gorenstein. In the 2nd part, we assume that $R$ is CM with a canonical module $ω$. For CM $R$-modules $M$ and $N$, we show that the vanishing of one of the following implies the same for others: ${\rm Ext}_R^{\gg 0}(M,N^{+})$, ${\rm Ext}_R^{\gg 0}(N,M^{+})$ and ${\rm Tor}_{\gg 0}^R(M,N)$, where $M^{+}$ denotes ${\rm Ext}_R^{d-\dim(M)}(M,ω)$. This strengthens a result of Huneke and Jorgensen. Furthermore, we prove a similar result for Tate cohomologies under the additional condition that $R$ is Gorenstein. |
| title | Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology |
| topic | Commutative Algebra 13D05, 13D07, 13H10 |
| url | https://arxiv.org/abs/2302.06267 |