The Stacks project

Example 22.33.6. Let $R$ be a ring. Let $(A, \text{d}) \to (B, \text{d})$ be a homomorphism of differential graded $R$-algebras. Then we can view $B$ as a differential graded $(A, B)$-bimodule and we get a functor

\[ - \otimes _ A B : D(A, \text{d}) \longrightarrow D(B, \text{d}) \]

By Lemma 22.33.5 the left adjoint of this is the functor $R\mathop{\mathrm{Hom}}\nolimits (B, -)$. For a differential graded $B$-module let us denote $N_ A$ the differential graded $A$-module obtained from $N$ by restriction via $A \to B$. Then we clearly have a canonical isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(B, N) \longrightarrow N_ A,\quad f \longmapsto f(1) \]

functorial in the $B$-module $N$. Thus we see that $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ is the restriction functor and we obtain

\[ \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(M, N_ A) = \mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(M \otimes ^\mathbf {L}_ A B, N) \]

bifunctorially in $M$ and $N$ exactly as in the case of commutative rings. Finally, observe that restriction is a tensor functor as well, since $N_ A = N \otimes _ B {}_ BB_ A = N \otimes _ B^\mathbf {L} {}_ BB_ A$ where ${}_ BB_ A$ is $B$ viewed as a differential graded $(B, A)$-bimodule.

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