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The étale-open topology and the stable fields conjecture

Will Johnson, Chieu-Minh Tran, Erik Walsberg, Jinhe Ye

Abstract

For an arbitrary field $K$ and $K$-variety $V$, we introduce the étale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably closed, real closed, or $p$-adically closed, respectively. Topological properties of the étale-open topology corresponds to algebraic properties of $K$. For example, the étale-open topology on $\mathbb{A}^1(K)$ is not discrete if and only if $K$ is large. As an application, we show that a large stable field is separably closed.

The étale-open topology and the stable fields conjecture

Abstract

For an arbitrary field and -variety , we introduce the étale-open topology on the set of -points of . This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when is separably closed, real closed, or -adically closed, respectively. Topological properties of the étale-open topology corresponds to algebraic properties of . For example, the étale-open topology on is not discrete if and only if is large. As an application, we show that a large stable field is separably closed.

Paper Structure

This paper contains 32 sections, 65 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Algebro-geometric preliminaries
  5. Model-theoretic preliminaries
  6. Direct proof of Theorem D
  7. General facts on systems of topologies
  8. Reduction to affine varieties
  9. The Zariski and discrete systems
  10. Field topologies
  11. Hausdorffness and disconnectedness
  12. Restriction and extension of systems
  13. Restriction
  14. Restriction of Zariski and field topologies
  15. Extension
  16. ...and 17 more sections

Key Result

Corollary 2.10

Suppose that $\delta(x,y)$ is stable and invariant, and $G$ is $\delta$-connected. Then there is a unique generic type $p \in S_\delta(G)$ and any $X \in \mathrm{Def}_\delta(G)$ is generic if and only if $p$ concentrates on $X$. If $X \in \mathrm{Def}_\delta(G)$ then exactly one of $X$ or $G \setmin

Theorems & Definitions (131)

  • Remark 1.1
  • Definition 1.2
  • proof
  • proof
  • Corollary 2.10
  • Proposition 2.11
  • proof
  • Theorem 3.1
  • Lemma 3.3
  • proof
  • ...and 121 more