The Stacks project

Lemma 69.13.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$-module. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Denote $Z \subset X$ the corresponding closed subspace and set $U = X \setminus Z$. There is a canonical isomorphism

\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n\mathcal{G}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}|_ U, \mathcal{F}|_ U). \]

In particular we have an isomorphism

\[ \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}^ n, \mathcal{F}) \longrightarrow \Gamma (U, \mathcal{F}). \]

Proof. Let $W$ be an affine scheme and let $W \to X$ be a surjective ├ętale morphism (see Properties of Spaces, Lemma 66.6.3). Set $R = W \times _ X W$. Then $W$ and $R$ are Noetherian schemes, see Morphisms of Spaces, Lemma 67.23.5. Hence the result hold for the restrictions of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{I}$, $U$, $Z$ to $W$ and $R$ by Cohomology of Schemes, Lemma 30.10.5. It follows formally that the result holds over $X$. $\square$

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