The Stacks project

Lemma 77.4.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent sheaf on $X$ which is flat over $Y$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Then we have

\[ \text{WeakAss}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) = \text{Ass}_{X/Y}(\mathcal{F}) \cap |f|^{-1}(\text{WeakAss}_ Y(\mathcal{G})) \]

Proof. Immediate consequence of Lemma 77.4.3. $\square$

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