Obstructions to curvature of modules over Cohen-Macaulay rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909915415576576 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with residue field $k$. If $M$ is a finitely generated $A$-module then set $\text{curv}(M) = \limsup_n\sqrt[n]{β_n^A(M)}$. We show that under mild hypotheses the existence of a single module $M$ with $1 \leq \text{curv}(M) < \text{curv}(k)$ imposes obstructions to both $\text{curv}(k)$ and $\text{curv}(M)$. Similarly we show that the condition $\text{Tor}^A_n(M, N) = 0$ for $n \gg 0$ imposes constraints on both $\text{curv}(M)$ and $\text{curv}(N)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_16109 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Obstructions to curvature of modules over Cohen-Macaulay rings Puthenpurakal, Tony J. Commutative Algebra Primary 13D02, Secondary 13D40, 13H10 Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with residue field $k$. If $M$ is a finitely generated $A$-module then set $\text{curv}(M) = \limsup_n\sqrt[n]{β_n^A(M)}$. We show that under mild hypotheses the existence of a single module $M$ with $1 \leq \text{curv}(M) < \text{curv}(k)$ imposes obstructions to both $\text{curv}(k)$ and $\text{curv}(M)$. Similarly we show that the condition $\text{Tor}^A_n(M, N) = 0$ for $n \gg 0$ imposes constraints on both $\text{curv}(M)$ and $\text{curv}(N)$. |
| title | Obstructions to curvature of modules over Cohen-Macaulay rings |
| topic | Commutative Algebra Primary 13D02, Secondary 13D40, 13H10 |
| url | https://arxiv.org/abs/2511.16109 |