The Stacks project

Lemma 52.15.3. Let $X$ be a Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. The functor

\[ \textit{Coh}(X, \mathcal{I}) \longrightarrow \text{Pro-}\mathit{QCoh}(\mathcal{O}_ X) \]

is fully faithful, see Categories, Remark 4.22.5.

Proof. Let $(\mathcal{F}_ n)$ and $(\mathcal{G}_ n)$ be objects of $\textit{Coh}(X, \mathcal{I})$. A morphism of pro-objects $\alpha $ from $(\mathcal{F}_ n)$ to $(\mathcal{G}_ n)$ is given by a system of maps $\alpha _ n : \mathcal{F}_{n'(n)} \to \mathcal{G}_ n$ where $\mathbf{N} \to \mathbf{N}$, $n \mapsto n'(n)$ is an increasing function. Since $\mathcal{F}_ n = \mathcal{F}_{n'(n)}/\mathcal{I}^ n\mathcal{F}_{n'(n)}$ and since $\mathcal{G}_ n$ is annihilated by $\mathcal{I}^ n$ we see that $\alpha _ n$ induces a map $\mathcal{F}_ n \to \mathcal{G}_ n$. $\square$

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