The fundamental fiber sequence in \'etale homotopy theory

TL;DR

This paper provides a conceptual proof that certain étale homotopy sequences are fiber sequences, explaining the relationship between the étale homotopy groups of a scheme and its geometric fiber, using profinite Galois categories.

Contribution

It offers a quick, conceptual proof of fiber sequences in étale homotopy theory using profinite Galois categories, clarifying the structure of étale homotopy groups.

Findings

Higher étale homotopy groups of a scheme and its geometric fiber are isomorphic.

The sequence of profinite étale fundamental groups is exact when the geometric fiber is connected.

Analogous results are established for the 'groupe fondamental élargi' of SGA3.

Abstract

Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be a qcqs $k$-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences \begin{equation*} \Pi_{<\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to \Pi_{<\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \qquad \text{and} \qquad \widehat{\Pi}{}_{\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to \widehat{\Pi}{}_{\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \end{equation*} of protruncated and profinite \'etale homotopy types are fiber sequences. This gives a common conceptual reason for the following two phenomena: first, the higher \'etale homotopy groups of $X$ and the geometric fiber $X_{\bar{k}}$ are isomorphic, and second, if $X_{\bar{k}}$ is connected, then the sequence of profinite \'etale fundamental groups $1\to\hat{\pi}{}_{1}^{\mathrm{\acute{e}t}}(X_{\bar{k}})\to\hat{\pi}{}_{1}^{\mathrm{\acute{e}t}}(X)\to\mathrm{Gal}(\bar{k}/k)\to 1$ is exact. It also proves the analogous results for the `groupe fondamental \'elargi' of SGA3.