Proposition 58.27.2 (0ELD)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.27: Lefschetz for the fundamental group
Proposition 58.27.2 (cite)

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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. $\square$

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