Lemma 22.29.1. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Then $M \mapsto M \otimes _ A N$ defines a functor

\[ - \otimes _ A N : \text{Mod}^{dg}_{(A, \text{d})} \longrightarrow \text{Mod}^{dg}_{(B, \text{d})} \]

of differential graded categories. This functor induces functors

\[ \text{Mod}_{(A, \text{d})} \to \text{Mod}_{(B, \text{d})} \quad \text{and}\quad K(\text{Mod}_{(A, \text{d})}) \to K(\text{Mod}_{(B, \text{d})}) \]

by an application of Lemma 22.26.5.

**Proof.**
Above we have seen how the construction defines a functor of underlying graded categories. Thus it suffices to show that the construction is compatible with differentials. Let $M$ and $M'$ be differential graded $A$-modules and let $f : M \to M'$ be an $A$-module homomorphism which is homogeneous of degree $n$. Then we have

\[ \text{d}(f) = \text{d}_{M'} \circ f - (-1)^ n f \circ \text{d}_ M \]

On the other hand, we have

\[ \text{d}(f \otimes \text{id}_ N) = \text{d}_{M' \otimes _ A N} \circ (f \otimes \text{id}_ N) - (-1)^ n (f \otimes \text{id}_ N) \circ \text{d}_{M \otimes _ A N} \]

Applying this to an element $x \otimes y$ with $x \in M$ and $y \in N$ homogeneous we get

\begin{align*} \text{d}(f \otimes \text{id}_ N)(x \otimes y) = & \text{d}_{M'}(f(x)) \otimes y + (-1)^{n + \deg (x)}f(x) \otimes \text{d}_ N(y) \\ & - (-1)^ n f(\text{d}_ M(x)) \otimes y - (-1)^{n + \deg (x)}f(x) \otimes \text{d}_ N(y) \\ = & \text{d}(f) (x \otimes y) \end{align*}

Thus we see that $\text{d}(f) \otimes \text{id}_ N = \text{d}(f \otimes \text{id}_ N)$ and the proof is complete.
$\square$

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