Lemma 36.3.6 (08DV)—The Stacks project

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Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.3: Derived category of quasi-coherent modules
Lemma 36.3.6 (cite)

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Lemma 36.3.6. Let $X = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. If $K^\bullet$ is a K-flat complex of $A$ -modules, then $\widetilde{K^\bullet }$ is a K-flat complex of $\mathcal{O}_ X$ -modules.

Proof. By More on Algebra, Lemma 15.59.3 we see that $K^\bullet \otimes _ A A_\mathfrak p$ is a K-flat complex of $A_\mathfrak p$ -modules for every $\mathfrak p \in \mathop{\mathrm{Spec}}(A)$ . Hence we conclude from Cohomology, Lemma 20.26.4 (and Schemes, Lemma 26.5.4) that $\widetilde{K^\bullet }$ is K-flat. $\square$

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