Lemma 74.42.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$. Then

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

**Proof.**
Since $\mathcal{H}$ is a sheaf on $X_{\acute{e}tale}$ and since we have étale descent for objects of $\textit{Coh}(X, \mathcal{I})$ it suffices to prove this étale locally. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.5.
$\square$

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