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 ${\pi }_1^{\mathrm{et}}$ 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 $$1\to {\pi }_1^{\mathrm{proet}}(X_{\bar{k}})\to {\pi }_1^{\mathrm{proet}}(X)\to {\mathrm{Gal}}_k\to 1$$ is exact as abstract groups and the map ${\pi }_1^{\mathrm{proet}}(X_{\bar{k}})\to {\pi }_1^{\mathrm{proet}}(X)$ is a topological embedding. On the way, we prove a general van Kampen theorem and the Künneth formula for the pro-étale fundamental group.

Major revision. The title has slightly changed. The main proof has been completely rewritten -- simplified and improved -- to make the it more readable. A detailed sketch of an alternative, quick approach added as a final remark. Various smaller changes throughout the article. Some minor typos and errors corrected. The numbering has changed. Comments always welcome!

References in corpus (2)

• Submersions and effective descent of etale morphisms
• Fundamental groups of topological stacks with slice property