Fundamental Exact Sequence for the Pro-Étale Fundamental Group

• Published in: arXiv.org 2022-08
• Main Author: Lara, Marcin
• Format: Article
• Language: English
• Subjects: Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1910.14015

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 $$1 \rightarrow \pi_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow \pi_1^{\mathrm{proet}}(X) \rightarrow \mathrm{Gal}_k \rightarrow 1$$ is exact as abstract groups and the map \(\pi_1^{\mathrm{proet}}(X_{\bar{k}}) \rightarrow \pi_1^{\mathrm{proet}}(X)\) is a topological embedding. On the way, we prove a general van Kampen theorem and the K\"unneth formula for the pro-étale fundamental group.