Lemma 22.34.2. Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. Assume that (22.34.1.1) is an isomorphism. 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 a differential graded $(A, C)$-bimodule $N''$ described in the proof.
Proof.
By Lemma 22.33.3 we may replace $N$ and $N'$ by quasi-isomorphic bimodules. Thus we may assume $N$, resp. $N'$ has property (P) as differential graded $(A, B)$-bimodule, resp. $(B, C)$-bimodule, see Lemma 22.28.4. We claim the lemma holds with the $(A, C)$-bimodule $N'' = N \otimes _ B N'$. To prove this, it suffices to show that
\[ N_ B \otimes _ B^\mathbf {L} N' \longrightarrow (N \otimes _ B N')_ C \]
is an isomorphism in $D(C, \text{d})$, see Lemma 22.34.1.
Let $F_\bullet $ be the filtration on $N$ as in property (P) for bimodules. By Lemma 22.28.5 there is a short exact sequence
\[ 0 \to \bigoplus \nolimits F_ iN \to \bigoplus \nolimits F_ iN \to N \to 0 \]
of differential graded $(A, B)$-bimodules which is split as a sequence of graded $(A, B)$-bimodules. A fortiori this is an admissible short exact sequence of differential graded $B$-modules and this produces a distinguished triangle
\[ \bigoplus \nolimits F_ iN_ B \to \bigoplus \nolimits F_ iN_ B \to N_ B \to \bigoplus \nolimits F_ iN_ B[1] \]
in $D(B, \text{d})$. Using that $- \otimes _ B^\mathbf {L} N'$ is an exact functor of triangulated categories and commutes with direct sums and using that $- \otimes _ B N'$ transforms admissible exact sequences into admissible exact sequences and commutes with direct sums we reduce to proving that
\[ (F_ pN)_ B \otimes _ B^\mathbf {L} N' \longrightarrow (F_ pN)_ B \otimes _ B N' \]
is a quasi-isomorphism for all $p$. Repeating the argument with the short exact sequences of $(A, B)$-bimodules
\[ 0 \to F_ pN \to F_{p + 1}N \to F_{p + 1}N/F_ pN \to 0 \]
which are split as graded $(A, B)$-bimodules we reduce to showing the same statement for $F_{p + 1}N/F_ pN$. Since these modules are direct sums of shifts of $(A \otimes _ R B)_ B$ we reduce to showing that
\[ (A \otimes _ R B)_ B \otimes _ B^\mathbf {L} N' \longrightarrow (A \otimes _ R B)_ B \otimes _ B N' \]
is a quasi-isomorphism.
Choose a filtration $F_\bullet $ on $N'$ as in property (P) for bimodules. Choose a quasi-isomorphism $P \to (A \otimes _ R B)_ B$ of differential graded $B$-modules where $P$ has property (P). We have to show that $P \otimes _ B N' \to (A \otimes _ R B)_ B \otimes _ B N'$ is a quasi-isomorphism because $P \otimes _ B N'$ represents $(A \otimes _ R B)_ B \otimes _ B^\mathbf {L} N'$ in $D(C, \text{d})$ by the construction in Lemma 22.33.2. As $N' = \mathop{\mathrm{colim}}\nolimits F_ pN'$ we find that it suffices to show that $P \otimes _ B F_ pN' \to (A \otimes _ R B)_ B \otimes _ B F_ pN'$ is a quasi-isomorphism. Using the short exact sequences $0 \to F_ pN' \to F_{p + 1}N' \to F_{p + 1}N'/F_ pN' \to 0$ which are split as graded $(B, C)$-bimodules we reduce to showing $P \otimes _ B F_{p + 1}N'/F_ pN' \to (A \otimes _ R B)_ B \otimes _ B F_{p + 1}N'/F_ pN'$ is a quasi-isomorphism for all $p$. Then finally using that $F_{p + 1}N'/F_ pN'$ is a direct sum of shifts of ${}_ B(B \otimes _ R C)_ C$ we conclude that it suffices to show that
\[ P \otimes _ B {}_ B(B \otimes _ R C)_ C \to (A \otimes _ R B)_ B \otimes _ B {}_ B(B \otimes _ R C)_ C \]
is a quasi-isomorphism. Since $P \to (A \otimes _ R B)_ B$ is a resolution by a module satisfying property (P) this map of differential graded $C$-modules represents the morphism (22.34.1.1) in $D(C, \text{d})$ and the proof is complete.
$\square$
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