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On schemes with trivial higher étale homotopy groups

Mohammed Moutand

Abstract

Let $(X,\bar x)$ be a pointed connected noetherian scheme. In this note, we give characterizations for the vanishing of the second étale homotopy group $π^{\rm ét}_2(X,\bar x)$ in terms of splitting profinite-étale covers of $X$, and by means of universal covering spaces of the Artin-Mazur-Friedlander étale homotopy type $Et(X)$. In particular, this provides certain classes of schemes for which the Brauer map is surjective.

On schemes with trivial higher étale homotopy groups

Abstract

Let be a pointed connected noetherian scheme. In this note, we give characterizations for the vanishing of the second étale homotopy group in terms of splitting profinite-étale covers of , and by means of universal covering spaces of the Artin-Mazur-Friedlander étale homotopy type . In particular, this provides certain classes of schemes for which the Brauer map is surjective.
Paper Structure (12 sections, 11 theorems, 36 equations)

This paper contains 12 sections, 11 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Etale homotopy type
  4. Remarks on locally constant constructible sheaves
  5. Truncated spaces and Truncated morphisms
  6. Profinite-étale covers and universal covering spaces of $Et(X)$
  7. Proof of the main result
  8. Further comments and Examples
  9. Brauer groups and Brauer map
  10. Smooth curves and abelian varieties
  11. Schröer spaces
  12. The case of proper schemes

Key Result

Theorem 1.1

Let $X$ be a connected noetherian geometrically unibranch scheme with a geometric base point $\bar{x} \rightarrow X$. Let $p: \tilde{Et}(X) \rightarrow Et(X)$ be the universal covering space of the étale homotopy type $Et(X)$, and let $\hat{f}: \hat{X} \rightarrow X$ be the pro-universal cover of X,

Theorems & Definitions (17)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Corollary 3.4
  • Proposition 3.5
  • ...and 7 more