Lemma 15.59.4. Let $A \to B \to C$ be ring maps. Let $N^\bullet $ be a complex of $B$-modules and $K^\bullet $ a complex of $C$-modules. The compositions of the functors

\[ D(A) \xrightarrow {- \otimes _ A^\mathbf {L} N^\bullet } D(B) \xrightarrow {- \otimes _ B^\mathbf {L} K^\bullet } D(C) \]

is the functor $- \otimes _ A^\mathbf {L} (N^\bullet \otimes _ B^\mathbf {L} K^\bullet ) : D(A) \to D(C)$. If $M$, $N$, $K$ are modules over $A$, $B$, $C$, then we have

\[ (M \otimes _ A^\mathbf {L} N) \otimes _ B^\mathbf {L} K = M \otimes _ A^\mathbf {L} (N \otimes _ B^\mathbf {L} K) = (M \otimes _ A^\mathbf {L} C) \otimes _ C^\mathbf {L} (N \otimes _ B^\mathbf {L} K) \]

in $D(C)$. We also have a canonical isomorphism

\[ (M \otimes _ A^\mathbf {L} N) \otimes _ B^\mathbf {L} K \longrightarrow (M \otimes _ A^\mathbf {L} K) \otimes _ C^\mathbf {L} (N \otimes _ B^\mathbf {L} C) \]

using signs. Similar results holds for complexes.

**Proof.**
Choose a K-flat complex $P^\bullet $ of $B$-modules and a quasi-isomorphism $P^\bullet \to N^\bullet $ (Lemma 15.58.12). Let $M^\bullet $ be a K-flat complex of $A$-modules representing an arbitrary object of $D(A)$. Then we see that

\[ (M^\bullet \otimes _ A^\mathbf {L} P^\bullet ) \otimes _ B^\mathbf {L} K^\bullet \longrightarrow (M^\bullet \otimes _ A^\mathbf {L} N^\bullet ) \otimes _ B^\mathbf {L} K^\bullet \]

is an isomorphism by Lemma 15.59.2 applied to the material inside the brackets. By Lemmas 15.58.5 and 15.58.6 the complex

\[ \text{Tot}(M^\bullet \otimes _ A P^\bullet ) = \text{Tot}((M^\bullet \otimes _ R A) \otimes _ A P^\bullet \]

is K-flat as a complex of $B$-modules and it represents the derived tensor product in $D(B)$ by construction. Hence we see that $(M^\bullet \otimes _ A^\mathbf {L} P^\bullet ) \otimes _ B^\mathbf {L} K^\bullet $ is represented by the complex

\[ \text{Tot}(\text{Tot}(M^\bullet \otimes _ A P^\bullet )\otimes _ B K^\bullet ) = \text{Tot}(M^\bullet \otimes _ A \text{Tot}(P^\bullet \otimes _ B K^\bullet )) \]

of $C$-modules. Equality by Homology, Remark 12.18.4. Going back the way we came we see that this is equal to

\[ M^\bullet \otimes _ A^\mathbf {L} (P^\bullet \otimes _ B^\mathbf {L} K^\bullet ) \longleftarrow M^\bullet \otimes _ A^\mathbf {L} (N^\bullet \otimes _ B^\mathbf {L} K^\bullet ) \]

The arrow is an isomorphism by definition of the functor $-\otimes _ B^\mathbf {L} K^\bullet $. All of these constructions are functorial in the complex $M^\bullet $ and hence we obtain our isomorphism of functors.

By the above we have the first equality in

\[ (M \otimes _ A^\mathbf {L} N) \otimes _ B^\mathbf {L} K = M \otimes _ A^\mathbf {L} (N \otimes _ B^\mathbf {L} K) = (M \otimes _ A^\mathbf {L} C) \otimes _ C^\mathbf {L} (N \otimes _ B^\mathbf {L} K) \]

The second equality follows from the final statement of Lemma 15.59.1. The same thing allows us to write $N \otimes _ B^\mathbf {L} K = (N \otimes _ B^\mathbf {L} C) \otimes _ C^\mathbf {L} K$ and substituting we get

\begin{align*} (M \otimes _ A^\mathbf {L} N) \otimes _ B^\mathbf {L} K & = (M \otimes _ A^\mathbf {L} C) \otimes _ C^\mathbf {L} ((N \otimes _ B^\mathbf {L} C) \otimes _ C^\mathbf {L} K) \\ & = (M \otimes _ A^\mathbf {L} C) \otimes _ C^\mathbf {L} (K \otimes _ C^\mathbf {L} (N \otimes _ B^\mathbf {L} C)) \\ & = ((M \otimes _ A^\mathbf {L} C) \otimes _ C^\mathbf {L} K) \otimes _ C^\mathbf {L} (N \otimes _ B^\mathbf {L} C)) \\ & = (M \otimes _ C^\mathbf {L} K) \otimes _ C^\mathbf {L} (N \otimes _ B^\mathbf {L} C) \end{align*}

by Lemmas 15.58.16 and 15.58.17 as well as the previously mentioned lemma.
$\square$

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