The Stacks project

Lemma 30.24.1. Let $A$ be Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{I} = I\mathcal{O}_ X$. Then the functor (30.23.3.1) is fully faithful.

Proof. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Then $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$ is a coherent $\mathcal{O}_ X$-module, see Modules, Lemma 17.21.5. By Lemma 30.23.5 the map

\[ \mathop{\mathrm{lim}}\nolimits _ n H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) \to \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ) \]

is bijective. Hence fully faithfulness of (30.23.3.1) follows from the theorem on formal functions (Lemma 30.20.6) for the coherent sheaf $\mathcal{H}$. $\square$


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