The Stacks project

Lemma 52.15.4. Let $I \subset \mathfrak a$ be ideals of a Noetherian ring $A$. Let $U = \mathop{\mathrm{Spec}}(A) \setminus V(\mathfrak a)$. Assume

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

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

3. for $\mathfrak p \subset A$ with $\mathfrak p \not\in V(I)$ and $V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$ we have $\text{depth}(A_\mathfrak p) \geq 2$.

Then the completion functor

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