The Stacks project

Lemma 52.15.7. Let $I \subset \mathfrak a \subset A$ be ideals of a Noetherian ring $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(I)$ ordered by reverse inclusion. Assume

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

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

Then the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {V} \textit{Coh}(\mathcal{O}_ V) \longrightarrow \textit{Coh}(U, I\mathcal{O}_ U), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \]

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

Proof. By Lemma 52.15.2 it suffices to show that

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

This follows immediately from Proposition 52.12.3. $\square$

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