On Tor-vanishing of local rings
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| Format: | Preprint |
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2025
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| _version_ | 1866913914812366848 |
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| author | Shekhar, Shrikant Gupta, Anjan |
| author_facet | Shekhar, Shrikant Gupta, Anjan |
| contents | Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the Tor-vanishing property if the converse holds, that is, $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $pd_R(M) < \infty$ or $pd_R(N) < \infty$. Interest in the Tor-vanishing property stems from the fact that Cohen-Macaulay local rings satisfying this property also satisfy the Auslander-Reiten conjecture.
In this article, we study a variant of this property. If $R$ is a generalized Golod ring, we prove that $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $\{curv_R M, curv_R N \} \cap \{0, 1\} \neq \emptyset$. A key intermediate step in our proof is to show that $curv_R M \in \{0, 1, curv_R k\}$ for any module $M$ over a generalized Golod ring $R$. As an application, we prove that generic Gorenstein local rings, non-trivial connected sums of generalized Golod-Gorenstein rings satisfy the Tor-vanishing property and consequently the Auslander-Reiten conjecture. Our method suggests a uniform approach and recovers many old results on the Tor-vanishing property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21764 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Tor-vanishing of local rings Shekhar, Shrikant Gupta, Anjan Commutative Algebra 13D07, 13D02, 13H10, 13D40 Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the Tor-vanishing property if the converse holds, that is, $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $pd_R(M) < \infty$ or $pd_R(N) < \infty$. Interest in the Tor-vanishing property stems from the fact that Cohen-Macaulay local rings satisfying this property also satisfy the Auslander-Reiten conjecture. In this article, we study a variant of this property. If $R$ is a generalized Golod ring, we prove that $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $\{curv_R M, curv_R N \} \cap \{0, 1\} \neq \emptyset$. A key intermediate step in our proof is to show that $curv_R M \in \{0, 1, curv_R k\}$ for any module $M$ over a generalized Golod ring $R$. As an application, we prove that generic Gorenstein local rings, non-trivial connected sums of generalized Golod-Gorenstein rings satisfy the Tor-vanishing property and consequently the Auslander-Reiten conjecture. Our method suggests a uniform approach and recovers many old results on the Tor-vanishing property. |
| title | On Tor-vanishing of local rings |
| topic | Commutative Algebra 13D07, 13D02, 13H10, 13D40 |
| url | https://arxiv.org/abs/2506.21764 |