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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

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group π1 ét defined in SGA1 and the more general π1SGA3 . 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 π1 proét(X k¯) π1 proét(X) Gal k 1

is exact as abstract groups and the map π1 proét(Xk¯) π1 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.

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
Received: 27 January 2020
Revised: 13 February 2023
Accepted: 13 May 2023
Published: 26 February 2024
Marcin Lara
Institute of Mathematics
Goethe University Frankfurt
Institute of Mathematics
Faculty of Mathematics and Computer Science
Jagiellonian University, Kraków

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