Lemma 71.5.3 (0CZ6)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 71: Divisors on Algebraic Spaces
Section 71.5: Fitting ideals
Lemma 71.5.3 (cite)

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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

$Z \subset Z_0 \subset X$ 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

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

Proof. Recall that formation of $Z$ commutes with étale 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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