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Fundamental Exact Sequence for the Pro-Étale Fundamental Group

Marcin Lara

Abstract

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual étale fundamental group $π_1^{\mathrm{et}}$ defined in SGA1 and the more general group defined in SGA3. It controls local systems in the pro-étale topology and leads to an interesting class of "geometric covers" of schemes, generalizing finite étale covers. We prove the homotopy exact sequence over a field for the pro-étale fundamental group of a geometrically connected scheme $X$ of finite type over a field $k$, i.e. that the sequence $$1 \rightarrow π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X) \rightarrow \mathrm{Gal}_k \rightarrow 1$$ is exact as abstract groups and the map $π_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow π_1^{\mathrm{proet}}(X)$ is a topological embedding. On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group.

Fundamental Exact Sequence for the Pro-Étale Fundamental Group

Abstract

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual étale fundamental group defined in SGA1 and the more general group defined in SGA3. It controls local systems in the pro-étale topology and leads to an interesting class of "geometric covers" of schemes, generalizing finite étale covers. We prove the homotopy exact sequence over a field for the pro-étale fundamental group of a geometrically connected scheme of finite type over a field , i.e. that the sequence is exact as abstract groups and the map is a topological embedding. On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group.

Paper Structure

This paper contains 16 sections, 50 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Conventions and notations
  3. Infinite Galois categories, Noohi groups and $\pi_1^{\mathrm{pro\mathrm{\acute{e}t}}}$
  4. Overview of the results in BhattScholze
  5. Noohi completion
  6. Dictionary between Noohi groups and $G-\mathrm{Sets}$
  7. A remark on valuative criteria
  8. Seifert-van Kampen theorem for $\pi_1^{\mathrm{pro\mathrm{\acute{e}t}}}$ and its applications
  9. Abstract Seifert–van Kampen theorem for infinite Galois categories
  10. Application to the pro-étale fundamental group
  11. Künneth formula
  12. Invariance of $\pi_1^{\mathrm{pro\mathrm{\acute{e}t}}}$ of a proper scheme under a base-change $K \supset k$ of algebraically closed fields
  13. Fundamental exact sequence
  14. Statement of the results and examples
  15. Preparation for the proof of Theorem \ref{['injectivity-on-the-left']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $X$ be a geometrically connected scheme of finite type over a field $k$. Then the sequence is exact as abstract groups. Moreover, the map $\pi_1^{\mathrm{pro\mathrm{\acute{e}t}}}(X_{\bar{k}}) \rightarrow \pi_1^{\mathrm{pro\mathrm{\acute{e}t}}}(X)$ is a topological embedding and the map $\pi_1^{\mathrm{pro\mathrm{\acute{e}t}}}(X) \rightarrow \mathrm{Gal}_k$ is a quotient map of topological

Theorems & Definitions (125)

  • Theorem : Thm. \ref{['exactness-in-geometric-to-arithmetic-as-abstract']}
  • Theorem : van Kampen theorem, Cor. \ref{['geomvK']} + Rmk. \ref{['remark-relations-in-vK-for-normalization']} + Prop. \ref{['properdescent']}, cf. Stix
  • Proposition : Künneth formula for $\pi_1^{\mathrm{pro\mathrm{\acute{e}t}}}$, Prop. \ref{['Kunneth-proetale']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 115 more