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Affineness of the maximal étale locus

Ivan Zelich

Abstract

In this paper we will prove a strong version of the celebrated purity of the ramification locus theorem in algebraic geometry. Our key input is a Tor-independence result for global sections of étale schemes over excellent regular local rings, which we will prove by tilting to perfect rings.

Affineness of the maximal étale locus

Abstract

In this paper we will prove a strong version of the celebrated purity of the ramification locus theorem in algebraic geometry. Our key input is a Tor-independence result for global sections of étale schemes over excellent regular local rings, which we will prove by tilting to perfect rings.
Paper Structure (4 sections, 16 theorems, 13 equations)

This paper contains 4 sections, 16 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed characteristic
  4. Affineness of the maximal étale locus

Key Result

Theorem 1.1

Let $X \rightarrow Y$ be a morphism of locally of finite-type between locally Noetherian schemes. Let $x \in X$ be a point and let $y = f(x)$. Assume that $\mathcal{O}_{X,x}$ is normal, $\mathcal{O}_{Y,y}$ is regular, and $\dim \mathcal{O}_{X,x} = \dim \mathcal{O}_{Y,y}\ge 1$. If for every point spe

Theorems & Definitions (28)

  • Theorem 1.1: Purity of the ramification locus
  • Theorem 1.2: Affineness of the complement of the ramification locus, equicharacteristic case
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Affineness of the complement of the ramification locus
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 18 more