The Stacks project

Lemma 15.68.4. Let $R$ be a ring. Given $K, L, M$ in $D(R)$ there is a canonical morphism

\[ R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} R\mathop{\mathrm{Hom}}\nolimits _ R(K, L) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(K, M) \]

in $D(R)$ functorial in $K, L, M$.

Proof. Choose a K-injective complex $I^\bullet $ representing $M$, a K-injective complex $J^\bullet $ representing $L$, and any complex of $R$-modules $K^\bullet $ representing $K$. By Lemma 15.67.2 there is a map of complexes

\[ \text{Tot}\left( \mathop{\mathrm{Hom}}\nolimits ^\bullet (J^\bullet , I^\bullet ) \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , J^\bullet ) \right) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , I^\bullet ) \]

The complexes of $R$-modules $\mathop{\mathrm{Hom}}\nolimits ^\bullet (J^\bullet , I^\bullet )$, $\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , J^\bullet )$, and $\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , I^\bullet )$ represent $R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)$, $R\mathop{\mathrm{Hom}}\nolimits _ R(K, L)$, and $R\mathop{\mathrm{Hom}}\nolimits _ R(K, M)$. If we choose a K-flat complex $H^\bullet $ and a quasi-isomorphism $H^\bullet \to \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , J^\bullet )$, then there is a map

\[ \text{Tot}\left( \mathop{\mathrm{Hom}}\nolimits ^\bullet (J^\bullet , I^\bullet ) \otimes _ R H^\bullet \right) \longrightarrow \text{Tot}\left( \mathop{\mathrm{Hom}}\nolimits ^\bullet (J^\bullet , I^\bullet ) \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , J^\bullet ) \right) \]

whose source represents $R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} R\mathop{\mathrm{Hom}}\nolimits _ R(K, L)$. Composing the two displayed arrows gives the desired map. We omit the proof that the construction is functorial. $\square$


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