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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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