The Stacks project

58.27 Lefschetz for the fundamental group

Of course we have already proven a bunch of results of this type in the local case. In this section we discuss the projective case.

Proposition 58.27.1. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Assume that for all $x \in X \setminus Y$ we have

\[ \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) > 1 \]

Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is fully faithful. In fact, for any open subscheme $V \subset X$ containing $Y$ the restriction functor $\textit{FÉt}_ V \to \textit{FÉt}_ Y$ is fully faithful.

Proof. The first statement is a formal consequence of Lemma 58.17.6 and Algebraic and Formal Geometry, Proposition 52.28.1. The second statement follows from Lemma 58.17.6 and Algebraic and Formal Geometry, Lemma 52.28.2. $\square$

Proposition 58.27.2. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Let $\mathcal{V}$ be the set of open subschemes of $X$ containing $Y$ ordered by reverse inclusion. Assume that for all $x \in X \setminus Y$ we have

\[ \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) > 2 \]

Then the restriction functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{FÉt}_ V \to \textit{FÉt}_ Y \]

is an equivalence.

Proof. This is a formal consequence of Lemma 58.17.4 and Algebraic and Formal Geometry, Proposition 52.28.7. $\square$

Proposition 58.27.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Assume that for all $x \in X \setminus Y$ we have

\[ \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) > 2 \]

and that for $x \in X \setminus Y$ closed purity holds for $\mathcal{O}_{X, x}$. Then the restriction functor $\textit{FÉt}_ X \to \textit{FÉt}_ Y$ is an equivalence. If $X$ or equivalently $Y$ is connected, then

\[ \pi _1(Y, \overline{y}) \to \pi _1(X, \overline{y}) \]

is an isomorphism for any geometric point $\overline{y}$ of $Y$.

Proof. Fully faithfulness holds by Proposition 58.27.1. By Proposition 58.27.2 any object of $\textit{FÉt}_ Y$ is isomorphic to the fibre product $U \times _ V Y$ for some finite étale morphism $U \to V$ where $V \subset X$ is an open subscheme containing $Y$. The complement $T = X \setminus V$ is1 a finite set of closed points of $X \setminus Y$. Say $T = \{ x_1, \ldots , x_ n\} $. By assumption we can find finite étale morphisms $V'_ i \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i})$ agreeing with $U \to V$ over $V \times _ X \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i})$. By Limits, Lemma 32.19.1 applied $n$ times we see that $U \to V$ extends to a finite étale morphism $U' \to X$ as desired. See Lemma 58.8.1 for the final statement. $\square$

[1] Namely, $T$ is proper over $k$ (being closed in $X$) and affine (being closed in the affine scheme $X \setminus Y$, see Morphisms, Lemma 29.43.18) and hence finite over $k$ (Morphisms, Lemma 29.44.11). Thus $T$ is a finite set of closed points.

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