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 32.20.1 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
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