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The étale open topology over the fraction field of a henselian local domain

Will Johnson, Erik Walsberg, Jinhe Ye

Abstract

Suppose that $R$ is a local domain with fraction field $K$. If $R$ is Henselian then the $R$-adic topology over $K$ refines the étale open topology. If $R$ is regular then the étale open topology over $K$ refines the $R$-adic topology. In particular the étale open topology over $L((t_1,\ldots,t_n))$ agrees with the $L[[t_1,\ldots,t_n]]$-adic topology for any field $L$ and $n \ge 1$.

The étale open topology over the fraction field of a henselian local domain

Abstract

Suppose that is a local domain with fraction field . If is Henselian then the -adic topology over refines the étale open topology. If is regular then the étale open topology over refines the -adic topology. In particular the étale open topology over agrees with the -adic topology for any field and .

Paper Structure

This paper contains 7 sections, 17 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Conventions and background
  3. R-adic topologies and V-topologies
  4. Proof of Theorem 1.2.1
  5. Proof of Theorem 1.2.2
  6. Proof of Theorem 1.3
  7. Final remark

Key Result

Theorem 1.2

Suppose that $R$ is a local domain with fraction field $K$ and $V$ is a $K$-variety. Hence the étale open topology over the fraction field $L((t_1,\ldots,t_n))$ of $L[[t_1,\ldots,t_n]]$ agrees with the $L[[t_1,\ldots,t_n]]$-adic topology.

Theorems & Definitions (34)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 1.7
  • proof
  • Proposition 2.1
  • Lemma 2.2
  • proof : Proof (of Lemma \ref{['lem:2-val']})
  • proof
  • proof
  • ...and 24 more