The Stacks project

Lemma 71.4.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Assume

  1. $\mathcal{F}$ is flat over $Y$,

  2. $X$ and $Y$ are locally Noetherian, and

  3. the fibres of $f$ are locally Noetherian.

Then

\[ \text{Ass}_ X(\mathcal{F} \otimes _{\mathcal{O}_ X} f^*\mathcal{G}) = \{ x \in \text{Ass}_{X/Y}(\mathcal{F})\text{ such that } f(x) \in \text{Ass}_ Y(\mathcal{G}) \} \]

Proof. Via étale localization, this is an immediate consequence of the result for schemes, see Divisors, Lemma 31.3.1. The result for schemes is more general only because we haven't defined associated points for non-Noetherian algebraic spaces (hence we need to assume $X$ and the fibres of $X \to Y$ are locally Noetherian to even be able to formulate this result). $\square$

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