The Stacks project

Lemma 58.6.3. Let $K$ be a field and set $X = \mathop{\mathrm{Spec}}(K)$. Let $\overline{K}$ be an algebraic closure and denote $\overline{x} : \mathop{\mathrm{Spec}}(\overline{K}) \to X$ the corresponding geometric point. Let $K^{sep} \subset \overline{K}$ be the separable algebraic closure.

  1. The functor of Lemma 58.2.2 induces an equivalence

    \[ \textit{FÉt}_ X \longrightarrow \textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets}. \]

    compatible with $F_{\overline{x}}$ and the functor $\textit{Finite-}\text{Gal}(K^{sep}/K)\textit{-Sets} \to \textit{Sets}$.

  2. This induces a canonical isomorphism

    \[ \text{Gal}(K^{sep}/K) \longrightarrow \pi _1(X, \overline{x}) \]

    of profinite topological groups.

Proof. The functor of Lemma 58.2.2 is the same as the functor $F_{\overline{x}}$ because for any $Y$ étale over $X$ we have

\[ \mathop{\mathrm{Mor}}\nolimits _ X(\mathop{\mathrm{Spec}}(\overline{K}), Y) = \mathop{\mathrm{Mor}}\nolimits _ X(\mathop{\mathrm{Spec}}(K^{sep}), Y) \]

Namely, as seen in the proof of Lemma 58.2.2 we have $Y = \coprod _{i \in I} \mathop{\mathrm{Spec}}(L_ i)$ with $L_ i/K$ finite separable over $K$. Hence any $K$-algebra homomorphism $L_ i \to \overline{K}$ factors through $K^{sep}$. Also, note that $F_{\overline{x}}(Y)$ is finite if and only if $I$ is finite if and only if $Y \to X$ is finite étale. This proves (1).

Part (2) is a formal consequence of (1), Lemma 58.3.11, and Lemma 58.3.3. (Please also see the remark below.) $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 58.6: Fundamental groups

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BNE. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 58.6.3 as pdf