Lemma 15.58.5. Let $R \to R'$ be a ring map. If $K^\bullet $ is a K-flat complex of $R$-modules, then $K^\bullet \otimes _ R R'$ is a K-flat complex of $R'$-modules.

**Proof.**
Follows from the definitions and the fact that $(K^\bullet \otimes _ R R') \otimes _{R'} L^\bullet = K^\bullet \otimes _ R L^\bullet $ for any complex $L^\bullet $ of $R'$-modules.
$\square$

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