Lemma 52.15.4. Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

1. $A$ is $I$-adically complete and has a dualizing complex,

2. for any associated prime $\mathfrak p \subset A$, $I \not\subset \mathfrak p$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ we have $\dim ((A/\mathfrak p)_\mathfrak q) > \text{cd}(A, I) + 1$.

3. for $\mathfrak p \subset A$, $I \not\subset \mathfrak p$ with with $V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$ we have $\text{depth}(A_\mathfrak p) \geq 2$.

Then the completion functor

$\textit{Coh}(\mathcal{O}_ U) \longrightarrow \textit{Coh}(U, I\mathcal{O}_ U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge$

is fully faithful on the full subcategory of finite locally free objects.

Proof. By Lemma 52.15.1 it suffices to show that

$\Gamma (U, \mathcal{O}_ U) = \mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{O}_ U/I^ n\mathcal{O}_ U)$

This follows immediately from Lemma 52.12.3. $\square$

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