Proposition 52.28.7. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module and let $s \in \Gamma (X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$ and denote $\mathcal{I} \subset \mathcal{O}_ X$ the corresponding sheaf of ideals. 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 completion functor
\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(X, \mathcal{I}) \]
is an equivalence on the full subcategories of finite locally free objects.
Proof.
To prove fully faithfulness it suffices to prove that
\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \Gamma (V, \mathcal{L}^{\otimes m}) \longrightarrow \mathop{\mathrm{lim}}\nolimits \Gamma (Y_ n, \mathcal{L}^{\otimes m}|_{Y_ n}) \]
is an isomorphism for all $m$, see Lemma 52.15.2. This follows from Lemma 52.28.2.
Essential surjectivity. Let $(\mathcal{F}_ n)$ be a finite locally free object of $\textit{Coh}(X, \mathcal{I})$. Then for $y \in Y$ we have $\mathcal{F}_ y^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_{n, y}$ is is a finite free $\mathcal{O}_{X, y}^\wedge $-module. Let $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ be a prime with $\mathfrak p \not\in V(\mathcal{I}_ y^\wedge )$. Then $\mathfrak p$ lies over a prime $\mathfrak p_0 \subset \mathcal{O}_{X, y}$ which corresponds to a specialization $x \leadsto y$ with $x \not\in Y$. By Local Cohomology, Lemma 51.11.3 and some dimension theory (see Varieties, Section 33.20) we have
\[ \text{depth}((\mathcal{O}_{X, y}^\wedge )_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) = \text{depth}(\mathcal{O}_{X, x}) + \dim (\overline{\{ x\} }) - \dim (\overline{\{ y\} }) \]
Thus our assumptions imply the assumptions of Proposition 52.28.5 are satisfied and we find that $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ X$-module $\mathcal{F}$. It then follows that $\mathcal{F}_ y$ is finite free for all $y \in Y$ and hence $\mathcal{F}$ is finite locally free in an open neighbourhood $V$ of $Y$. This finishes the proof.
$\square$
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