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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