Let $R$ be a ring. The derived hom we will define in this section is a functor
This is an internal hom in the derived category of $R$-modules in the sense that it is characterized by the formula
for objects $K, L, M$ of $D(R)$. Note that this formula characterizes the objects up to unique isomorphism by the Yoneda lemma. A construction can be given as follows. Choose a K-injective complex $I^\bullet $ of $R$-modules representing $M$, choose a complex $L^\bullet $ representing $L$, and set
with notation as in Section 15.71. A generalization of this construction is discussed in Differential Graded Algebra, Section 22.31. From (15.71.0.1) and Derived Categories, Lemma 13.31.2 that we have
for all $n \in \mathbf{Z}$. In particular, the object $R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)$ of $D(R)$ is well defined, i.e., independent of the choice of the K-injective complex $I^\bullet $.
Lemma 15.73.1. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical isomorphism in $D(R)$ functorial in $K, L, M$ which recovers (15.73.0.1) by taking $H^0$.
\[ R\mathop{\mathrm{Hom}}\nolimits _ R(K, R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)) = R\mathop{\mathrm{Hom}}\nolimits _ R(K \otimes _ R^\mathbf {L} L, M) \]
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
by Lemma 15.71.1. 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. $\square$
Lemma 15.73.2. 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 (15.71.0.1) 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
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. $\square$
Lemma 15.73.3. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism in $D(R)$ functorial in $K, L, M$.
\[ R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(R\mathop{\mathrm{Hom}}\nolimits _ R(K, L), M) \]
Proof. Choose a K-injective complex $I^\bullet $ representing $M$, a K-injective complex $J^\bullet $ representing $L$, and a K-flat complex $K^\bullet $ representing $K$. The map is defined using the map
of Lemma 15.71.6. We omit the proof that this is functorial in all three objects of $D(R)$. $\square$
Lemma 15.73.4. Let $R$ be a ring. Given $K, L, M$ in $D(R)$ there is a canonical morphism in $D(R)$ functorial in $K, L, M$.
\[ R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) \otimes _ R^\mathbf {L} R\mathop{\mathrm{Hom}}\nolimits _ R(K, L) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(K, M) \]
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 15.71.3 there is a map of complexes
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
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. $\square$
Lemma 15.73.5. Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$ there is a canonical morphism in $D(R)$ functorial in $K$, $L$, $M$.
\[ K \otimes _ R^\mathbf {L} R\mathop{\mathrm{Hom}}\nolimits _ R(M, L) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(M, K \otimes _ R^\mathbf {L} L) \]
Proof. Choose a K-flat complex $K^\bullet $ representing $K$, and a K-injective complex $I^\bullet $ representing $L$, and choose any complex $M^\bullet $ representing $M$. Choose a quasi-isomorphism $\text{Tot}(K^\bullet \otimes _ R I^\bullet ) \to J^\bullet $ where $J^\bullet $ is K-injective. Then we use the map
where the first map is the map from Lemma 15.71.4. $\square$
Lemma 15.73.6. Let $R$ be a ring. Given complexes $K, L$ in $D(R)$ there is a canonical morphism in $D(R)$ functorial in both $K$ and $L$.
\[ K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ R(L, K \otimes _ R^\mathbf {L} L) \]
Proof. This is a special case of Lemma 15.73.5 but we will also prove it directly. Choose a K-flat complex $K^\bullet $ representing $K$ and any complex $L^\bullet $ representing $L$. Choose a quasi-isomorphism $\text{Tot}(K^\bullet \otimes _ R L^\bullet ) \to J^\bullet $ where $J^\bullet $ is K-injective. Then we use the map
where the first map is the map from Lemma 15.71.5. $\square$
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