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Fields of Fractions in Rigid Geometry

Jiahong Yu

TL;DR

The paper shows that in rigid geometry, the normalization of an affinoid integral K-algebra can be reconstructed from its field of fractions by intersecting all maximal discrete valuation rings, enabling a fully faithful link between normal affinoid domains and field extensions of K. This provides a p-adic analogue of the Riemann Hebbarkeitssatz and yields practical criteria for finiteness of morphisms, along with applications to prismatic cohomology and dagger algebras. The approach hinges on valuation-theoretic reconstruction and Nagata/excellence properties of affinoid algebras, bridging birational-like data with analytic structure in the non-Archimedean setting.

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.

Fields of Fractions in Rigid Geometry

TL;DR

The paper shows that in rigid geometry, the normalization of an affinoid integral K-algebra can be reconstructed from its field of fractions by intersecting all maximal discrete valuation rings, enabling a fully faithful link between normal affinoid domains and field extensions of K. This provides a p-adic analogue of the Riemann Hebbarkeitssatz and yields practical criteria for finiteness of morphisms, along with applications to prismatic cohomology and dagger algebras. The approach hinges on valuation-theoretic reconstruction and Nagata/excellence properties of affinoid algebras, bridging birational-like data with analytic structure in the non-Archimedean setting.

Abstract

Let be an affinoid integral domain over a non-Archimedean field , and let be its field of fractions. We prove that the normalization of can be reconstructed from 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 to the category of field extensions of . This provides another -adic analogue of the Riemann Hebbarkeitssatz.
Paper Structure (3 sections, 17 theorems, 15 equations)

This paper contains 3 sections, 17 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. First properties of affinoid algebras
  3. Proof of the main theorems

Key Result

Theorem 1.1

Let $A$ be an affinoid $K$-algebra whose underlying ring is an integral domain. Let $\mathcal{E}$ be the field of fractions of $A$ and $B \subseteq \mathcal{E}$ be the normalization of $A$. Denote by $\mathbb{X}(\mathcal{E})$ the set of all discrete valuations of $\mathcal{E}$. Then where $O_v$ is the valuation ring of $v$, and $B^\circ$ is the subring of power-bounded elements of $B$.

Theorems & Definitions (34)

  • Theorem 1.1: Theorem \ref{['theo:reconst2']}
  • Theorem 1.2: Theorem \ref{['theo: proved ration implies algebra']}
  • Theorem 1.3: Theorem \ref{['cor: max affinoid of finitely generated field proof']}
  • Theorem 1.4: guo2021prismaticcohomologyrigidanalytic, Theorem \ref{['prismatic proof']}
  • Definition 2.1: Bosch_2014
  • Theorem 2.2
  • proof
  • Definition 2.3: Bosch_2014
  • Theorem 2.4
  • proof
  • ...and 24 more