Lemma 75.21.4. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $\mathcal{G}^\bullet $ be a bounded above complex of quasi-coherent $\mathcal{O}_ X$-modules flat over $Y$. Then formation of
commutes with arbitrary base change (see proof for precise statement).
Proof. The statement means the following. Let $g : Y' \to Y$ be a morphism of algebraic spaces and consider the base change diagram
in other words $X' = Y' \times _ Y X$. The lemma asserts that
is an isomorphism. Observe that on the right hand side we do not use derived pullback on $\mathcal{G}^\bullet $. To prove this, we apply Lemmas 75.21.2 and 75.21.3 to see that it suffices to prove the canonical map
satisfies the equivalent conditions of Lemma 75.21.1. This follows by checking the condition on stalks, where it immediately follows from the fact that $\mathcal{G}^\bullet _{\overline{x}} \otimes _{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{Y', \overline{y}'}$ computes the derived tensor product by our assumptions on the complex $\mathcal{G}^\bullet $. $\square$
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