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 $π_1^{\mathrm{et}}$ 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.