Homotopy Exact Sequence for the Pro-Étale Fundamental Group II

Abstract

The pro-étale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes the usual étale fundamental group ${\pi }_1^{\mathrm{et}}$ defined in SGA1 and leads to an interesting class of "geometric coverings" of schemes, generalizing finite étale covers. We prove exactness of the general homotopy sequence for the pro-étale fundamental group, i.e. that for a geometric point $\bar{s}$ on $S$ and a flat proper morphism $X\to S$ of finite presentation whose geometric fibres are connected and reduced, the sequence $${\pi }_1^{\mathrm{proet}}(X_{\bar{s}})\to {\pi }_1^{\mathrm{proet}}(X)\to {\pi }_1^{\mathrm{proet}}(S)\to 1$$ is "nearly exact". This generalizes a theorem of Grothendieck from finite étale covers to geometric coverings. We achieve the proof by constructing an infinite (i.e. non-quasi-compact) analogue of the Stein factorization in this setting.

21 pages, comments welcome!

References in corpus (2)

• Revêtements étales et groupe fondamental (SGA 1)
• Géométrie anabélienne tempérée