Fields of Fractions in Rigid Geometry

TL;DR

This paper demonstrates how the normalization of an affinoid integral domain over a non-Archimedean field can be reconstructed from its field of fractions, establishing a fully faithful functor and offering a p-adic analogue of the Riemann Hebbarkeitssatz.

Contribution

It introduces a method to recover the normalization from the field of fractions and shows the functorial relationship between normal affinoid domains and field extensions.

Findings

Normalization can be reconstructed from the field of fractions.

The functor from normal affinoid domains to field extensions is fully faithful.

Provides a p-adic analogue of the Riemann Hebbarkeitssatz.

Abstract

Let $A$ be an affinoid integral domain over a non-Archimedean field $K$, and let $L$ be its field of fractions. We prove that the normalization of $A$ can be reconstructed from $L$ by taking the intersection of all maximal discrete valuation subrings. As a corollary, taking the field of fractions induces a fully faithful functor from the category of normal affinoid integral domains over $K$ to the category of field extensions of $K$. This provides another $p$-adic analogue of the Riemann Hebbarkeitssatz.