Lemma 67.15.2 (07TZ)—The Stacks project

Lemma 67.15.2. 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. Then

The support of $\mathcal{F}$ is closed.
For a geometric point $\overline{x}$ lying over $x \in |X|$ we have

$x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_{\overline{x}} \not= 0 \Leftrightarrow \mathcal{F}_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}} \kappa (\overline{x}) \not= 0.$
For any morphism of algebraic spaces $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$ .

Proof. Choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ . By Lemma 67.15.1 the inverse image of the support of $\mathcal{F}$ is the support of $\varphi ^*\mathcal{F}$ which is closed by Morphisms, Lemma 29.5.3. Thus (1) follows from the definition of the topology on $|X|$ .

The first equivalence in (2) is the definition of support. The second equivalence follows from Nakayama's lemma, see Algebra, Lemma 10.20.1.

Let $f : Y \to X$ be as in (3). Note that $f^*\mathcal{F}$ is of finite type by Properties of Spaces, Section 66.30. For the final assertion, let $\overline{y}$ be a geometric point of $Y$ mapping to the geometric point $\overline{x}$ on $X$ . Recall that

$(f^*\mathcal{F})_{\overline{y}} = \mathcal{F}_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}} \mathcal{O}_{Y, \overline{y}},$

see Properties of Spaces, Lemma 66.29.5. Hence $(f^*\mathcal{F})_{\overline{y}} \otimes \kappa (\overline{y})$ is nonzero if and only if $\mathcal{F}_{\overline{x}} \otimes \kappa (\overline{x})$ is nonzero. By (2) this implies $x \in \text{Supp}(\mathcal{F})$ if and only if $y \in \text{Supp}(f^*\mathcal{F})$ , which is the content of assertion (3). $\square$

The Stacks project

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