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 (22.34.1.1) 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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