The Stacks project

Lemma 22.38.2. Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. Assume $A \otimes _ R C$ represents $A \otimes ^\mathbf {L}_ R C$ in $D(R)$. Let $N$ be a differential graded $(A, B)$-bimodule. Let $N'$ be a differential graded $(B, C)$-bimodule. Then the composition

\[ \xymatrix{ D(A, \text{d}) \ar[rr]^{- \otimes _ A^\mathbf {L} N} & & D(B, \text{d}) \ar[rr]^{- \otimes _ B^\mathbf {L} N'} & & D(C, \text{d}) } \]

is isomorphic to $- \otimes _ A^\mathbf {L} N''$ for some differential graded $(A, C)$-bimodule $N''$.

Proof. Using Lemma 22.38.1 we choose a quasi-isomorphism $(B', \text{d}) \to (B, \text{d})$ with $B'$ K-flat as a complex of $R$-modules. By Lemma 22.37.1 the functor $-\otimes ^\mathbf {L}_{B'} B : D(B', \text{d}) \to D(B, \text{d})$ is an equivalence with quasi-inverse given by restriction. Note that restriction is canonically isomorphic to the functor $- \otimes ^\mathbf {L}_ B B : D(B, \text{d}) \to D(B', \text{d})$ where $B$ is viewed as a $(B, B')$-bimodule. Thus it suffices to prove the lemma for the compositions

\[ D(A) \to D(B) \to D(B'),\quad D(B') \to D(B) \to D(C),\quad D(A) \to D(B') \to D(C). \]

The first one is Lemma 22.34.3 because $B'$ is K-flat as a complex of $R$-modules. The second one is true because $B \otimes _ B^\mathbf {L} N' = N' = B \otimes _ B N'$ and hence Lemma 22.34.1 applies. Thus we reduce to the case where $B$ is K-flat as a complex of $R$-modules.

Assume $B$ is K-flat as a complex of $R$-modules. It suffices to show that ( is an isomorphism, see Lemma 22.34.2. Choose a quasi-isomorphism $L \to A$ where $L$ is a differential graded $R$-module which has property (P). Then it is clear that $P = L \otimes _ R B$ has property (P) as a differential graded $B$-module. Hence we have to show that $P \to A \otimes _ R B$ induces a quasi-isomorphism

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

We can rewrite this as

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

Since $B$ is K-flat as a complex of $R$-modules, it follows from More on Algebra, Lemma 15.59.2 that it is enough to show that

\[ P \otimes _ R C \to A \otimes _ R C \]

is a quasi-isomorphism, which is exactly our assumption. $\square$

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