A note concerning the vanishing of local cohomology for roots in mixed characteristic
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908996204494848 |
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| author | Sridhar, Prashanth |
| author_facet | Sridhar, Prashanth |
| contents | The goal of this note is to record the following curious fact: let $(S,\n)$ be an unramified regular local ring of mixed characteristic $p>0$ and dimension $d$. Let $L$ denote the quotient field of $S$ and $K=L(ω)$ with $ω^p\in L$. Let $R$ denote the integral closure of $S$ in $K$. Then $R$ is Cohen-Macaulay if and only if $\mathrm{H}^{d-1}_{\n}(R)=0$, i.e., the obstruction to the Cohen-Macaulayness of $R$ lies in a single local cohomology module. Furthermore, this is equivalent to the dual module $\Hom_S(R,S)$ satisfying Serre's condition $(S_3)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note concerning the vanishing of local cohomology for roots in mixed characteristic Sridhar, Prashanth Commutative Algebra Algebraic Geometry 13H05 The goal of this note is to record the following curious fact: let $(S,\n)$ be an unramified regular local ring of mixed characteristic $p>0$ and dimension $d$. Let $L$ denote the quotient field of $S$ and $K=L(ω)$ with $ω^p\in L$. Let $R$ denote the integral closure of $S$ in $K$. Then $R$ is Cohen-Macaulay if and only if $\mathrm{H}^{d-1}_{\n}(R)=0$, i.e., the obstruction to the Cohen-Macaulayness of $R$ lies in a single local cohomology module. Furthermore, this is equivalent to the dual module $\Hom_S(R,S)$ satisfying Serre's condition $(S_3)$. |
| title | A note concerning the vanishing of local cohomology for roots in mixed characteristic |
| topic | Commutative Algebra Algebraic Geometry 13H05 |
| url | https://arxiv.org/abs/2506.09072 |