The Stacks project

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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