Lemma 15.68.1. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical isomorphism

$R\mathop{\mathrm{Hom}}\nolimits _ R(K, R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)) = R\mathop{\mathrm{Hom}}\nolimits _ R(K \otimes _ R^\mathbf {L} L, M)$

in $D(R)$ functorial in $K, L, M$ which recovers (15.68.0.1) by taking $H^0$.

Proof. Choose a K-injective complex $I^\bullet$ representing $M$ and a K-flat complex of $R$-modules $L^\bullet$ representing $L$. For any complex of $R$-modules $K^\bullet$ we have

$\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , I^\bullet )) = \mathop{\mathrm{Hom}}\nolimits ^\bullet (\text{Tot}(K^\bullet \otimes _ R L^\bullet ), I^\bullet )$

by Lemma 15.67.1. The lemma follows by the definition of $R\mathop{\mathrm{Hom}}\nolimits$ and because $\text{Tot}(K^\bullet \otimes _ R L^\bullet )$ represents the derived tensor product. $\square$

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