The Stacks project

Lemma 15.57.10. Let $R$ be a ring. Let $K_1^\bullet \to K_2^\bullet \to \ldots $ be a system of K-flat complexes. Then $\mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet $ is K-flat. More generally any filtered colimit of K-flat complexes is K-flat.

Proof. Because we are taking termwise colimits we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \text{Tot}(M^\bullet \otimes _ R K_ i^\bullet ) = \text{Tot}(M^\bullet \otimes _ R \mathop{\mathrm{colim}}\nolimits _ i K_ i^\bullet ) \]

by Algebra, Lemma 10.11.9. Hence the lemma follows from the fact that filtered colimits are exact, see Algebra, Lemma 10.8.8. $\square$

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