Lemma 56.5.6 (0FZG)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 56: Functors and Morphisms
Section 56.5: Functors between categories of quasi-coherent modules
Lemma 56.5.6 (cite)

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Lemma 56.5.6. In Lemma 56.5.2 or in Lemma 56.5.5 if $F$ is an exact functor, then the corresponding object $\mathcal{K}$ of $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$ is flat over $X$ .

Proof. We may assume $X$ is affine, so we are in the case of Lemma 56.5.2. By Lemma 56.5.4 we may assume $Y$ is affine. In the affine case the statement translates into Remark 56.3.5. $\square$

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