The Stacks project

Lemma 47.13.8. In Situation 47.13.6 assume that

  1. $E$ viewed as an object of $D(R)$ is compact, and

  2. $N = \mathop{\mathrm{Hom}}\nolimits ^\bullet _ R(E^\bullet , R)$ computes $R\mathop{\mathrm{Hom}}\nolimits (E, R)$.

Then $R\mathop{\mathrm{Hom}}\nolimits (E, -) : D(R) \to D(E)$ is isomorphic to $K \mapsto K \otimes _ R^\mathbf {L} N$.

Proof. Special case of Differential Graded Algebra, Lemma 22.33.9. $\square$

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