Lemma 52.24.2 (0EKZ)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 52: Algebraic and Formal Geometry
Section 52.24: Application to the completion functor
Lemma 52.24.2 (cite)

previous lemma

Lemma 52.24.2. In Situation 52.16.1 assume

$I = (f)$ is principal,
$A$ is $f$ -adically complete,
$f$ is a nonzerodivisor,
$H^1_\mathfrak a(A/fA)$ 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. $\square$

The Stacks project

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