The Stacks project

Lemma 81.10.8. In Situation 81.10.6. Let $X' \to X$ be a flat morphism of algebraic spaces. Set $Z' = X' \times _ X Z$ and $Y' = X' \times _ X Y$. The pullbacks $\mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_{X'})$ and $\mathit{QCoh}(Y \to X, Z) \to \mathit{QCoh}(Y' \to X', Z')$ are compatible with the functors (81.10.6.2) and 81.10.6.1).

Proof. This is true because pullback commutes with pullback and because flat pullback commutes with pushforward along quasi-compact and quasi-separated morphisms, see Cohomology of Spaces, Lemma 69.11.2. $\square$


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