The Fundamental Fiber Sequence in Étale Homotopy Theory

Abstract

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 $ \Pi _{<\infty }^{\'{e}\textrm {t}}(X_{\bar {k}}) \to \Pi _{<\infty }^{\'{e}\textrm {t}}(X) \to \text {BGal}(\bar {k}/k) $ and $ \widehat {\Pi }_{\infty }^{\'{e}\textrm {t}}(X_{\bar {k}}) \to \widehat {\Pi }_{\infty }^{\'{e}\textrm {t}}(X) \to \text {BGal}(\bar {k}/k) $ 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 {\pi }^{\'{e}\textrm {t}}_1(X_{\bar {k}}) \to \hat {\pi }^{\'{e}\textrm {t}}_1(X) \to \text {Gal}(\bar {k}/k) \to 1 $ is exact. It also proves the analogous results for the groupe fondamental élargi of SGA3.