The Stacks project

Theorem 58.6.2. Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$.

  1. The fibre functor $F_{\overline{x}}$ defines an equivalence of categories

    \[ \textit{FÉt}_ X \longrightarrow \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} \]
  2. Given a second geometric point $\overline{x}'$ of $X$ there exists an isomorphism $t : F_{\overline{x}} \to F_{\overline{x}'}$. This gives an isomorphism $\pi _1(X, \overline{x}) \to \pi _1(X, \overline{x}')$ compatible with the equivalences in (1). This isomorphism is independent of $t$ up to inner conjugation.

  3. Given a morphism $f : X \to Y$ of connected schemes denote $\overline{y} = f \circ \overline{x}$. There is a canonical continuous homomorphism

    \[ f_* : \pi _1(X, \overline{x}) \to \pi _1(Y, \overline{y}) \]

    such that the diagram

    \[ \xymatrix{ \textit{FÉt}_ Y \ar[r]_{\text{base change}} \ar[d]_{F_{\overline{y}}} & \textit{FÉt}_ X \ar[d]^{F_{\overline{x}}} \\ \textit{Finite-}\pi _1(Y, \overline{y})\textit{-Sets} \ar[r]^{f_*} & \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} } \]

    is commutative.

Proof. Part (1) follows from Lemma 58.5.5 and Proposition 58.3.10. Part (2) is a special case of Lemma 58.3.11. For part (3) observe that the diagram

\[ \xymatrix{ \textit{FÉt}_ Y \ar[r] \ar[d]_{F_{\overline{y}}} & \textit{FÉt}_ X \ar[d]^{F_{\overline{x}}} \\ \textit{Sets} \ar@{=}[r] & \textit{Sets} } \]

is commutative (actually commutative, not just $2$-commutative) because $\overline{y} = f \circ \overline{x}$. Hence we can apply Lemma 58.3.11 with the implied transformation of functors to get (3). $\square$

Comments (2)

Comment #8403 by Hayama Kazuma on

The "" that appears in the commutative diagram is more like "", since it is induced by taking base change of covers, and is already the map of étale fundamental groups.

Comment #9017 by on

Hahaha! Yes, of course. The notation is that is the map between groups and that the functor between the categories of sets with group actions is supposed to be induced by . See Lemma 58.3.11. Going to leave as is for now.

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

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