The Stacks project

Lemma 47.13.7. In Situation 47.13.6 the functor $R\mathop{\mathrm{Hom}}\nolimits (A, -)$ is equal to the composition of $R\mathop{\mathrm{Hom}}\nolimits (E, -) : D(R) \to D(E, \text{d})$ and the equivalence $- \otimes ^\mathbf {L}_ E A : D(E, \text{d}) \to D(A)$.

Proof. This is true because $R\mathop{\mathrm{Hom}}\nolimits (E, -)$ is the right adjoint to $- \otimes ^\mathbf {L}_ R E$, see Differential Graded Algebra, Lemma 22.33.5. Hence this functor plays the same role as the functor $R\mathop{\mathrm{Hom}}\nolimits (A, -)$ for the map $R \to A$ (Lemma 47.13.1), whence these functors must correspond via the equivalence $- \otimes ^\mathbf {L}_ E A : D(E, \text{d}) \to D(A)$. $\square$

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