Lemma 22.30.1. 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. The construction above defines a functor
\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, -) : \text{Mod}^{dg}_{(B, \text{d})} \longrightarrow \text{Mod}^{dg}_{(A, \text{d})}
of differential graded categories. This functor induces functors
\text{Mod}_{(B, \text{d})} \to \text{Mod}_{(A, \text{d})} \quad \text{and}\quad K(\text{Mod}_{(B, \text{d})}) \to K(\text{Mod}_{(A, \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 N_1 and N_2 be differential graded B-modules. Write
H_{12} = \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N_1, N_2),\quad H_1 = \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, N_1),\quad H_2 = \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, N_2)
Consider the composition
c : H_{12} \otimes _ R H_1 \longrightarrow H_2
in the differential graded category \text{Mod}^{dg}_{(B, \text{d})}. Let f : N_1 \to N_2 be a B-module homomorphism which is homogeneous of degree n, in other words, f \in H_{12}^ n. The functor in the lemma sends f to c_ f : H_1 \to H_2, g \mapsto c(f, g). Similarly for \text{d}(f). On the other hand, the differential on
\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(A, \text{d})}}(H_1, H_2)
sends c_ f to \text{d}_{H_2} \circ c_ f - (-1)^ n c_ f \circ \text{d}_{H_1}. As c is a morphism of complexes of R-modules we have \text{d} c(f, g) = c(\text{d}f, g) + (-1)^ n c(f, \text{d}g). Hence we see that
\begin{align*} (\text{d}c_ f)(g) & = \text{d}c(f,g) - (-1)^ n c(f, \text{d}g) \\ & = c(\text{d}f, g) + (-1)^ n c(f, \text{d}g) - (-1)^ n c(f, \text{d}g) \\ & = c(\text{d}f, g) = c_{\text{d}f}(g) \end{align*}
and the proof is complete.
\square
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