On the depth of tensor products over Cohen-Macaulay rings
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| Format: | Preprint |
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2025
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| _version_ | 1866908344613076992 |
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| author | Kimura, Kaito Lyle, Justin Soto-Levins, Andrew J. |
| author_facet | Kimura, Kaito Lyle, Justin Soto-Levins, Andrew J. |
| contents | Inspired by classical work on the depth formula for tensor products of finitely generated $R$-modules, we introduce two conditions which we call $(\mathbf{ldep})$ and $(\mathbf{rdep})$ and their derived variations. We show for Cohen-Macaulay local rings that derived $(\mathbf{ldep})$ is equivalent to $\dim(R)$ being a uniform Auslander bound for $R$, and if $\dim(R)>0$ that both are equivalent to $(\mathbf{ldep})$. We introduce an analogous condition we call the \emph{uniform Buchweitz condition} and provide a corresponding theorem for the $(\mathbf{rdep})$ condition. As a consequence of these results, we show $(\mathbf{ldep})$ implies $(\mathbf{rdep})$ when $R$ is Gorenstein and that the $(\mathbf{ldep})$ and $(\mathbf{rdep})$ conditions behave well under modding out by regular sequences and completion, but we give a concrete example showing they need not localize. Using our methods, we extend work of Jorgensen by calculating the value $q_R(M,N):=\sup\{i \mid \operatorname{Tor}^R_i(M,N) \ne 0\}$ under certain conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00441 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the depth of tensor products over Cohen-Macaulay rings Kimura, Kaito Lyle, Justin Soto-Levins, Andrew J. Commutative Algebra Primary 13C15, Secondary 13D07 Inspired by classical work on the depth formula for tensor products of finitely generated $R$-modules, we introduce two conditions which we call $(\mathbf{ldep})$ and $(\mathbf{rdep})$ and their derived variations. We show for Cohen-Macaulay local rings that derived $(\mathbf{ldep})$ is equivalent to $\dim(R)$ being a uniform Auslander bound for $R$, and if $\dim(R)>0$ that both are equivalent to $(\mathbf{ldep})$. We introduce an analogous condition we call the \emph{uniform Buchweitz condition} and provide a corresponding theorem for the $(\mathbf{rdep})$ condition. As a consequence of these results, we show $(\mathbf{ldep})$ implies $(\mathbf{rdep})$ when $R$ is Gorenstein and that the $(\mathbf{ldep})$ and $(\mathbf{rdep})$ conditions behave well under modding out by regular sequences and completion, but we give a concrete example showing they need not localize. Using our methods, we extend work of Jorgensen by calculating the value $q_R(M,N):=\sup\{i \mid \operatorname{Tor}^R_i(M,N) \ne 0\}$ under certain conditions. |
| title | On the depth of tensor products over Cohen-Macaulay rings |
| topic | Commutative Algebra Primary 13C15, Secondary 13D07 |
| url | https://arxiv.org/abs/2505.00441 |