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Pseudo-absolute values: foundations

Antoine Sédillot

Abstract

In this article, we introduce pseudo-absolute values, which generalise usual absolute values. Roughly speaking, a pseudo-absolute value on a field $K$ is a map $|\cdot| : K \to [0,+\infty]$ satisfying axioms similar to those of usual absolute values. This notion allows to include "pathological" absolute values one can encounter trying to incorporate the analogy between Diophantine approximation and Nevanlinna theory in an Arakelov theoretic framework. It turns out that the space of all pseudo-absolute values can be endowed with a compact Hausdorff topology in a similar way as the Berkovich analytic spectrum of a Banach ring. Moreover, we introduce both local and global notions of analytic spaces over pseudo-valued fields and interpret them as analytic counterparts to Zariski-Riemann spaces.

Pseudo-absolute values: foundations

Abstract

In this article, we introduce pseudo-absolute values, which generalise usual absolute values. Roughly speaking, a pseudo-absolute value on a field is a map satisfying axioms similar to those of usual absolute values. This notion allows to include "pathological" absolute values one can encounter trying to incorporate the analogy between Diophantine approximation and Nevanlinna theory in an Arakelov theoretic framework. It turns out that the space of all pseudo-absolute values can be endowed with a compact Hausdorff topology in a similar way as the Berkovich analytic spectrum of a Banach ring. Moreover, we introduce both local and global notions of analytic spaces over pseudo-valued fields and interpret them as analytic counterparts to Zariski-Riemann spaces.

Paper Structure

This paper contains 55 sections, 72 theorems, 119 equations.

Table of Contents

  1. Preliminaries
  2. Valuation rings and Prüfer domains
  3. Rank, rational rank, composite valuations and Gauss valuations
  4. Connection to algebraic geometry : specialisation on a scheme
  5. Prüfer domains
  6. Application to rings of analytic functions
  7. Models over a Prüfer domain
  8. Zariski-Riemann spaces
  9. Berkovich spaces
  10. Banach rings
  11. Analytification in the sense of Berkovich: completely valued field case
  12. Analytification in the sense of Berkovich: global case
  13. Pseudo-absolute values
  14. Definitions
  15. Example of pseudo-absolute values
  16. ...and 40 more sections

Key Result

Theorem A

Let $K$ be a field with prime subring $k$. Denote by $M_{K}$ the set of all pseudo-absolute values on $K$. Then the topological space $M_K$ is non-empty, compact and Hausdorff. Moreover, we have a specification map $j : M_K \to \operatorname{ZR}(K/k)$ which is continuous, where $\operatorname{ZR}(K/

Theorems & Definitions (183)

  • Remark
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1.1.1
  • Example 1.1.2
  • Example 1.1.3: Gauss valuations
  • Proposition 1.1.4: stacks, https://stacks.math.columbia.edu/tag/00PH
  • Proposition 1.1.5
  • proof
  • ...and 173 more