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Height reduction for local uniformization of varieties and non-archimedean spaces

Michael Temkin

Abstract

It is known since the works of Zariski that the essential difficulty in the local uniformization problem is met already in the case of valuations of height one. In this paper we prove that local uniformization of schemes and non-archimedean analytic spaces rigorously follows from the case of valuations of height one. For non-archimedean spaces this result reduces the problem to studying local structure of smooth Berkovich spaces.

Height reduction for local uniformization of varieties and non-archimedean spaces

Abstract

It is known since the works of Zariski that the essential difficulty in the local uniformization problem is met already in the case of valuations of height one. In this paper we prove that local uniformization of schemes and non-archimedean analytic spaces rigorously follows from the case of valuations of height one. For non-archimedean spaces this result reduces the problem to studying local structure of smooth Berkovich spaces.
Paper Structure (47 sections, 19 theorems, 15 equations)

This paper contains 47 sections, 19 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. History and motivation
  3. Global conjectures
  4. Local conjectures
  5. Main results
  6. Induction on height
  7. Overview of the paper
  8. The method
  9. Conventiones
  10. Local uniformization for schemes
  11. Local uniformization conjectures
  12. Valuations on integral schemes
  13. Noetherian case
  14. Regular pairs
  15. Resolution along a valuation
  16. ...and 32 more sections

Key Result

Lemma 2.1.3

If ${\lambda}\colon{\rm Spec}(R)\to X$ is a valuation on a noetherian scheme $X$ such that ${\rm Frac}(R)=k(X)$, then the valuation ring $R$ is of finite height $h$. In fact, $h\le d$, where $d={\rm dim}({\mathcal{O}}_{X,x})$ and $x$ is the center of ${\lambda}$.

Theorems & Definitions (55)

  • Lemma 2.1.3
  • proof
  • Definition 2.1.6
  • Remark 2.1.7
  • Definition 2.1.9
  • Conjecture 2.1.10
  • Remark 2.1.11
  • Lemma 2.2.2
  • proof
  • Lemma 2.2.4
  • ...and 45 more