Fundamental exact sequence for the pro-étale fundamental group

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups — the usual étale fundamental group ${\pi }_1^{\text{ ét}}$ defined in SGA1 and the more general ${\pi }_1^{\mathrm{SGA3}<!--FUNCTION\; APPLICATION-->}$. 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

$$1\to {\pi }_1^{\text{ proét}}(X_{\overline{k}})\to {\pi }_1^{\text{ proét}}(X)\to {\mathrm{Gal}<!--FUNCTION\; APPLICATION-->}_k\to 1$$

is exact as abstract groups and the map ${\pi }_1^{\text{ proét}}(X_{\overline{k}})\to {\pi }_1^{\text{ proét}}(X)$ is a topological embedding.

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