The fundamental fiber sequence in étale 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*} Î _{<\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to Î _{<\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \qquad \text{and} \qquad \widehatÎ {}_{\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to \widehatÎ {}_{\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \end{equation*} of protruncated and profinite étale homotopy types are fiber sequences. This gives a common conceptual reason for the following two phenomena: first, the higher étale 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 étale fundamental groups $1\to \hat{\ddot{I}}€{}_1^{\acute{\mathrm{e}}\mathrm{t}}(X_{\bar{k}})\to \hat{\ddot{I}}€{}_1^{\acute{\mathrm{e}}\mathrm{t}}(X)\to \mathrm{Gal}(\bar{k}/k)\to 1$ is exact. It also proves the analogous results for the `groupe fondamental élargi' of SGA3.

Comments very welcome! v3. 16 pages. Improved the exposition in subsection 1.2. Expanded and generalized the material in subsection 3.3. To appear in International Mathematics Research Notices. v2: 15 pages. Minor changes and added a reference. v1: 14 pages