On complete integral closedness of the $p$-adic completion of absolute integral closure

TL;DR

This paper investigates the complete integral closedness of the $p$-adic completion of the absolute integral closure of a Noetherian complete local domain in mixed characteristic, revealing conditions under which it is or isn't completely integrally closed.

Contribution

It proves that the $p$-adic completion of the absolute integral closure is completely integrally closed in an extended field but not in its own fraction field for dimensions two or higher.

Findings

$oxed{ ext{Complete integral closedness in an extended field} }$

$oxed{ ext{Not completely integrally closed in its own fraction field when } ext{dim}(R) ext{ } ext{geq } 2$

$oxed{ ext{Extension of integral closure properties in mixed characteristic} }$

Abstract

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 $\dim(R)\geq 2$.