The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Theorem 53.6.2. 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 53.5.5 and Proposition 53.3.10. Part (2) is a special case of Lemma 53.3.11. 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 53.3.11 with the implied transformation of functors to get (3). $\square$


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