Lemma 15.68.2. Let $R$ be a ring. Let $P^\bullet$ be a bounded above complex of projective $R$-modules. Let $L^\bullet$ be a complex of $R$-modules. Then $R\mathop{\mathrm{Hom}}\nolimits _ R(P^\bullet , L^\bullet )$ is represented by the complex $\mathop{\mathrm{Hom}}\nolimits ^\bullet (P^\bullet , L^\bullet )$.

Proof. By (15.67.0.1) and Derived Categories, Lemma 13.19.8 the cohomology groups of the complex are “correct”. Hence if we choose a quasi-isomorphism $L^\bullet \to I^\bullet$ with $I^\bullet$ a K-injective complex of $R$-modules then the induced map

$\mathop{\mathrm{Hom}}\nolimits ^\bullet (P^\bullet , L^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (P^\bullet , I^\bullet )$

is a quasi-isomorphism. As the right hand side is our definition of $R\mathop{\mathrm{Hom}}\nolimits _ R(P^\bullet , L^\bullet )$ we win. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).