The Stacks project

Lemma 52.15.5. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

  1. $A$ is $f$-adically complete,

  2. $H^1_\mathfrak a(A)$ and $H^2_\mathfrak a(A)$ are annihilated by a power of $f$.

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 52.12.7. $\square$

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