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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