The Stacks project

Lemma 15.58.9. 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 12.25.3 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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