Lemma 52.15.9. Let $A$ be a Noetherian ring. Let $f \in \mathfrak a \subset A$ be an element of an ideal of $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Let $\mathcal{V}$ be the set of open subschemes of $U$ containing $U \cap V(f)$ ordered by reverse inclusion. Assume

$A$ is $f$-adically complete,

$f$ is a nonzerodivisor,

$H^1_\mathfrak a(A/fA)$ is a finite $A$-module.

Then the completion functor

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

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