The Stacks project

Lemma 37.25.4. Let $f : X \to S$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $U \subset X$ be an open subscheme. Let $g : S' \to S$ be a morphism of schemes, let $f' : X' = X_{S'} \to S'$ be the base change of $f$, let $g' : X' \to X$ be the projection, set $\mathcal{F}' = (g')^*\mathcal{F}$, and set $U' = (g')^{-1}(U)$. Finally, let $s' \in S'$ with image $s = g(s')$. In this case

\[ \text{Ass}_{X_ s}(\mathcal{F}_ s) \subset U_ s \Leftrightarrow \text{Ass}_{X'_{s'}}(\mathcal{F}'_{s'}) \subset U'_{s'}. \]

Proof. This follows immediately from Divisors, Lemma 31.7.3. See also Divisors, Remark 31.7.4. $\square$

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