Lemma 22.31.4. Let (A, \text{d}) and (B, \text{d}) be differential graded algebras over a ring R. Let N be a differential graded (A, B)-bimodule. Then for every n \in \mathbf{Z} there are isomorphisms
H^ n(R\mathop{\mathrm{Hom}}\nolimits (N, M)) = \mathop{\mathrm{Ext}}\nolimits ^ n_{D(B, \text{d})}(N, M)
of R-modules functorial in M. It is also functorial in N with respect to the operation described in Lemma 22.31.3.
Proof.
In the proof of Lemma 22.31.2 we have seen
R\mathop{\mathrm{Hom}}\nolimits (N, M) = \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, I)
as a differential graded A-module where M \to I is a quasi-isomorphism of M into a differential graded B-module with property (I). Hence this complex has the correct cohomology modules by Lemma 22.22.3. We omit a discussion of the functorial nature of these identifications.
\square
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