Lemma 22.34.3 (09S4)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.34: Composition of derived tensor products
Lemma 22.34.3 (cite)

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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 15.59.2. $\square$

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