The Stacks project

Lemma 48.12.4. Let $g : Y' \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $f : X \to Y$ be a proper, flat morphism of finite presentation. Then the base change map (48.5.0.1) is an isomorphism for all $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. By Lemma 48.12.2 formation of the functors $a$ and $a'$ commutes with restriction to opens of $Y$ and $Y'$. Hence we may assume $Y' \to Y$ is a morphism of affine schemes, see Remark 48.6.1. In this case the statement follows from Lemma 48.6.2. $\square$

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