Lemma 52.15.6. 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

$A$ is $f$-adically complete,

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

Then the completion functor

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

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