The Stacks project

Lemma 71.3.7. Let $S$ be a scheme. Let $f : X \to Y$ be an étale morphism of algebraic spaces. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $x \in |X|$ be a point with image $y \in |Y|$. Then

\[ x \in \text{WeakAss}(f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G}) \]

Proof. This is immediate from the definition of weakly associated points and in fact the corresponding lemma for the case of schemes (More on Flatness, Lemma 38.2.8) is the basis for our definition. $\square$

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