On complete integral closedness of the $p$-adic completion of absolute integral closure
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910009263128576 |
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| author | Heitmann, Raymond Ma, Linquan |
| author_facet | Heitmann, Raymond Ma, Linquan |
| contents | Fix a prime $p$ and let $(R,\mathfrak{m})$ be a Noetherian complete local domain of mixed characteristic $(0,p)$ with fraction field $K$. Let $R^+$ denote the absolute integral closure of $R$, which is the integral closure of $R$ in an algebraic closure $\overline{K}$ of $K$. The first author has shown that $\widehat{R^+}$, the $p$-adic completion of $R^+$, is an integral domain. In this paper, we prove that $\widehat{R^+}$ is completely integrally closed in $\widehat{R^+}\otimes_{R^+}\overline{K}$, but $\widehat{R^+}$ is not completely integrally closed in its own fraction field when $\dim(R)\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On complete integral closedness of the $p$-adic completion of absolute integral closure Heitmann, Raymond Ma, Linquan Commutative Algebra Fix a prime $p$ and let $(R,\mathfrak{m})$ be a Noetherian complete local domain of mixed characteristic $(0,p)$ with fraction field $K$. Let $R^+$ denote the absolute integral closure of $R$, which is the integral closure of $R$ in an algebraic closure $\overline{K}$ of $K$. The first author has shown that $\widehat{R^+}$, the $p$-adic completion of $R^+$, is an integral domain. In this paper, we prove that $\widehat{R^+}$ is completely integrally closed in $\widehat{R^+}\otimes_{R^+}\overline{K}$, but $\widehat{R^+}$ is not completely integrally closed in its own fraction field when $\dim(R)\geq 2$. |
| title | On complete integral closedness of the $p$-adic completion of absolute integral closure |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2506.19148 |