The fundamental fiber sequence in \'etale homotopy theory

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.