The Stacks project

Lemma 22.34.3. Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. If $C$ is K-flat as a complex of $R$-modules, then (22.34.1.1) is an isomorphism and the conclusion of Lemma 22.34.2 is valid.

Proof. Choose a quasi-isomorphism $P \to (A \otimes _ R B)_ B$ of differential graded $B$-modules, where $P$ has property (P). Then we have to show that

\[ P \otimes _ B (B \otimes _ R C) \longrightarrow (A \otimes _ R B) \otimes _ B (B \otimes _ R C) \]

is a quasi-isomorphism. Equivalently we are looking at

\[ P \otimes _ R C \longrightarrow A \otimes _ R B \otimes _ R C \]

This is a quasi-isomorphism if $C$ is K-flat as a complex of $R$-modules by More on Algebra, Lemma 15.57.4. $\square$


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