On complete integral closedness of the $p$-adic completion of absolute integral closure
Raymond Heitmann, Linquan Ma
TL;DR
This paper tackles when the $p$-adic completion $\widehat{R^+}$ of the absolute integral closure is completely integrally closed in various ambient rings in mixed characteristic. The authors leverage perfectoid techniques, balanced big Cohen–Macaulay algebras, and extended plus closures to establish that $\widehat{R^+}$ is completely integrally closed in $\widehat{R^+}\otimes_{R^+}\overline{K}$, while it is not in its own fraction field when $\dim(R)\ge 2$, and provide a precise $p$-adic-topology criterion for complete integral closedness in principal localizations $\widehat{R^+}[1/g]$. They prove a chain of equivalences linking $p$-adic separation/completeness of $\widehat{R^+}/g$, valuation bounds $v_Q(g)$ at height-one primes over $p$, and bounded $p$-power torsion, and they construct explicit $g$ exhibiting non-separation in higher dimension. The results clarify the ring-theoretic behavior of mixed-characteristic closures and illuminate the role of $p$-adic topology in complete integral closure, with implications for perfectoid techniques and birational geometry 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 $\dim(R)\geq 2$.