The Stacks project

Lemma 66.15.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $U$ be a scheme and let $\varphi : U \to X$ be an ├ętale morphism. Then

\[ \text{Supp}(\varphi ^*\mathcal{F}) = |\varphi |^{-1}(\text{Supp}(\mathcal{F})) \]

where the left hand side is the support of $\varphi ^*\mathcal{F}$ as a quasi-coherent module on the scheme $U$.

Proof. Let $u\in U$ be a (usual) point and let $\overline{x}$ be a geometric point lying over $u$. By Properties of Spaces, Lemma 65.29.4 we have $(\varphi ^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = \mathcal{F}_{\overline{x}}$. Since $\mathcal{O}_{U, u} \to \mathcal{O}_{X, \overline{x}}$ is the strict henselization by Properties of Spaces, Lemma 65.22.1 we see that it is faithfully flat (see More on Algebra, Lemma 15.45.1). Thus we see that $(\varphi ^*\mathcal{F})_ u = 0$ if and only if $\mathcal{F}_{\overline{x}} = 0$. This proves the lemma. $\square$

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