The Stacks project

Lemma 52.24.2. In Situation 52.16.1 assume

  1. $I = (f)$ is principal,

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

  3. $f$ is a nonzerodivisor,

  4. $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$ are finite $A$-modules.

Then with $U_0 = U \cap V(f)$ the completion functor

\[ \mathop{\mathrm{colim}}\nolimits _{U_0 \subset U' \subset U\text{ open}} \textit{Coh}(\mathcal{O}_{U'}) \longrightarrow \textit{Coh}(U, f\mathcal{O}_ U) \]

is an equivalence on the full subcategories of finite locally free objects.

Proof. The functor is fully faithful by Lemma 52.15.8. Essential surjectivity follows from Lemma 52.16.11. $\square$

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