Lemma 71.4.7. Let S be a scheme. Let
\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }
be a cartesian diagram of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module and set \mathcal{F}' = (g')^*\mathcal{F}. If f is locally of finite type, then
x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') \Rightarrow g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})
if x \in \text{Ass}_{X/Y}(\mathcal{F}), then given y' \in |Y'| with f(x) = g(y'), there exists an x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') with g'(x') = x and f'(x') = y'.
Proof.
This follows from the case of schemes by étale localization. We write out the details completely. Choose a scheme V and a surjective étale morphism V \to Y. Choose a scheme U and a surjective étale morphism U \to V \times _ Y X. Choose a scheme V' and a surjective étale morphism V' \to V \times _ Y Y'. Then U' = V' \times _ V U is a scheme and the morphism U' \to X' is surjective and étale.
Proof of (1). Choose u' \in U' mapping to x'. Denote v' \in V' the image of u'. Then x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') is equivalent to u' \in \text{Ass}(\mathcal{F}|_{U'_{v'}}) by definition (writing \text{Ass} instead of \text{WeakAss} makes sense as U'_{v'} is locally Noetherian). Applying Divisors, Lemma 31.7.3 we see that the image u \in U of u' is in \text{Ass}(\mathcal{F}|_{U_ v}) where v \in V is the image of u. This in turn means g'(x') \in \text{Ass}_{X/Y}(\mathcal{F}).
Proof of (2). Choose u \in U mapping to x. Denote v \in V the image of u. Then x \in \text{Ass}_{X/Y}(\mathcal{F}) is equivalent to u \in \text{Ass}(\mathcal{F}|_{U_ v}) by definition. Choose a point v' \in V' mapping to y' \in |Y'| and to v \in V (possible by Properties of Spaces, Lemma 66.4.3). Let t \in \mathop{\mathrm{Spec}}(\kappa (v') \otimes _{\kappa (v)} \kappa (u)) be a generic point of an irreducible component. Let u' \in U' be the image of t. Applying Divisors, Lemma 31.7.3 we see that u' \in \text{Ass}(\mathcal{F}'|_{U'_{v'}}). This in turn means x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') where x' \in |X'| is the image of u'.
\square
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