Fundamental exact sequence for the pro-étale fundamental group

Lara, Marcin

doi:10.2140/ant.2024.18.631

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Fundamental exact sequence for the pro-étale fundamental group

Marcin Lara

Vol. 18 (2024), No. 4, 631–683

DOI: 10.2140/ant.2024.18.631

Abstract

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group $π_{1}^{ét}$ defined in SGA1 and the more general $π_{1}^{SGA3}$ . It controls local systems in the pro-étale topology and leads to an interesting class of “geometric coverings” of schemes, generalizing finite étale coverings.

We prove exactness of the fundamental sequence 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 \to π_{1}^{proét} (X_{\bar{k}}) \to π_{1}^{proét} (X) \to {Gal}_{k} \to 1

is exact as abstract groups and the map $π_{1}^{proét} (X_{\bar{k}}) \to π_{1}^{proét} (X)$ is a topological embedding.

On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group.

Keywords

pro-étale topology, pro-étale fundamental group, étale fundamental group, homotopy exact sequence, fundamental exact sequence, Noohi groups

Mathematical Subject Classification 2010

Primary: 14F35

Secondary: 14D10, 14F20, 20E06

Milestones

Received: 27 January 2020

Revised: 13 February 2023

Accepted: 13 May 2023

Published: 26 February 2024

Authors

Marcin Lara
Institute of Mathematics Goethe University Frankfurt Frankfurt Germany	Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University, Kraków Poland

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Vol. 18, No. 4, 2024