
Lemma 15.57.9. Let $R$ be a ring. Let $P^\bullet$ be a bounded above complex of flat $R$-modules. Then $P^\bullet$ is K-flat.

Proof. Let $L^\bullet$ be an acyclic complex of $R$-modules. Let $\xi \in H^ n(\text{Tot}(L^\bullet \otimes _ R P^\bullet ))$. We have to show that $\xi = 0$. Since $\text{Tot}^ n(L^\bullet \otimes _ R P^\bullet )$ is a direct sum with terms $L^ a \otimes _ R P^ b$ we see that $\xi$ comes from an element in $H^ n(\text{Tot}(\tau _{\leq m}L^\bullet \otimes _ R P^\bullet ))$ for some $m \in \mathbf{Z}$. Since $\tau _{\leq m}L^\bullet$ is also acyclic we may replace $L^\bullet$ by $\tau _{\leq m}L^\bullet$. Hence we may assume that $L^\bullet$ is bounded above. In this case the spectral sequence of Homology, Lemma 12.22.6 has

${}'E_1^{p, q} = H^ p(L^\bullet \otimes _ R P^ q)$

which is zero as $P^ q$ is flat and $L^\bullet$ acyclic. Hence $H^*(\text{Tot}(L^\bullet \otimes _ R P^\bullet )) = 0$. $\square$

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