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The Stacks project

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


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