The Stacks project

Lemma 71.5.3. 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

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

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

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

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

Proof. Recall that formation of $Z$ commutes with ├ętale localization, see Morphisms of Spaces, Definition 67.15.4 (which uses Morphisms of Spaces, Lemma 67.15.3 to define $Z$). Hence (1) and (2) follow from the case of schemes, see Divisors, Lemma 31.9.3. To get $\mathcal{G}_0$ as in part (3) we can use that we have $\mathcal{G}$ on $Z$ as in Morphisms of Spaces, Lemma 67.15.3 and set $\mathcal{G}_0 = (Z \to Z_0)_*\mathcal{G}$. $\square$

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