A note on cohomological vanishing theorems
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arXiv
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| Format: | Preprint |
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2024
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| author | Asgharzadeh, Mohsen |
| author_facet | Asgharzadeh, Mohsen |
| contents | We study $cd(M,N):=\sup\{j:H^j_{m}(M,N)\neq0\}$, and we prove the following over $AB$-rings: $cd(M,N)<\infty$ iff $cd(M, N)\leq2 dim R$. For locally free over the punctured spectrum, we present the better bound, namely $cd(M, N)<\infty$ iff $cd(M, N)\leq dim R,$ and show this is sharp for maximal Cohen-Macaulay, and prove that this detects freeness of $M$. We present some explicit examples to compute $cd(M, N)$. Now, suppose $R$ is only Cohen-Macaulay and of prime characteristic equipped with the Frobenius map $φ$. We show for some $n\gg 0$ that $cd(^{φ_n}R,M)<\infty$ iff $id_R(M)<\infty.$ This presents some criteria on regularity. Also, some vanishing results on $Ext^i_R(^φR,-)$ are given, where $(-)\in\{R,^φR\}$. We determine conditions under which the vanishing $Ext^i_R(^φR,-)$ of restricted many $i$-th, implies the vanishing of all. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_14133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on cohomological vanishing theorems Asgharzadeh, Mohsen Commutative Algebra We study $cd(M,N):=\sup\{j:H^j_{m}(M,N)\neq0\}$, and we prove the following over $AB$-rings: $cd(M,N)<\infty$ iff $cd(M, N)\leq2 dim R$. For locally free over the punctured spectrum, we present the better bound, namely $cd(M, N)<\infty$ iff $cd(M, N)\leq dim R,$ and show this is sharp for maximal Cohen-Macaulay, and prove that this detects freeness of $M$. We present some explicit examples to compute $cd(M, N)$. Now, suppose $R$ is only Cohen-Macaulay and of prime characteristic equipped with the Frobenius map $φ$. We show for some $n\gg 0$ that $cd(^{φ_n}R,M)<\infty$ iff $id_R(M)<\infty.$ This presents some criteria on regularity. Also, some vanishing results on $Ext^i_R(^φR,-)$ are given, where $(-)\in\{R,^φR\}$. We determine conditions under which the vanishing $Ext^i_R(^φR,-)$ of restricted many $i$-th, implies the vanishing of all. |
| title | A note on cohomological vanishing theorems |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2401.14133 |