Lemma 70.5.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Let $Z_0 \subset X$ be the closed subspace cut out by $\text{Fit}_0(\mathcal{F})$. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Then

$Z \subset Z_0 \subset X$ as closed subspaces,

$|Z| = |Z_0| = \text{Supp}(\mathcal{F})$ as closed subsets of $|X|$,

there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module $\mathcal{G}_0$ with

\[ (Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}. \]

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