Lemma 48.13.4. Let $f : X \to Y$ be a perfect proper morphism of Noetherian schemes. Let $g : Y' \to Y$ be a morphism with $Y'$ Noetherian. If $X$ and $Y'$ are tor independent over $Y$, 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 48.13.2 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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