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.
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