## 15.73 Derived hom

Let $R$ be a ring. The derived hom we will define in this section is a functor

$D(R)^{opp} \times D(R) \longrightarrow D(R),\quad (K, L) \longmapsto R\mathop{\mathrm{Hom}}\nolimits _ R(K, L)$

This is an internal hom in the derived category of $R$-modules in the sense that it is characterized by the formula

15.73.0.1
\begin{equation} \label{more-algebra-equation-internal-hom} \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K \otimes _ R^\mathbf {L} L, M) \end{equation}

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

$R\mathop{\mathrm{Hom}}\nolimits _ R(L, M) = \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , I^\bullet )$

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

15.73.0.2
\begin{equation} \label{more-algebra-equation-h0-RHom} H^ n(R\mathop{\mathrm{Hom}}\nolimits _ R(L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(R)}(L, M[n]) \end{equation}

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

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

in $D(R)$ functorial in $K, L, M$ which recovers (15.73.0.1) 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 15.71.1. 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

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

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. $\square$

Lemma 15.73.3. Let $R$ be a ring. Let $K, L, M$ be objects of $D(R)$. There is a canonical morphism

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

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 a K-flat complex $K^\bullet$ representing $K$. The map is defined using the map

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

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

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

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 15.71.3 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. $\square$

Lemma 15.73.5. Let $R$ be a ring. Given complexes $K, L, M$ in $D(R)$ there is a canonical morphism

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

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

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

$\text{Tot}\left( K^\bullet \otimes _ R \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , I^\bullet ) \right) \to \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , \text{Tot}(K^\bullet \otimes _ R I^\bullet )) \to \mathop{\mathrm{Hom}}\nolimits ^\bullet (M^\bullet , J^\bullet )$

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

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

in $D(R)$ functorial in both $K$ and $L$.

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

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

where the first map is the map from Lemma 15.71.5. $\square$

Comment #7935 by Karl Schwede on

It would be nice to also point out the cases where things like Lemma 0A67 and Lemma 0BYN are isomorphisms (unless this is already done elsewhere that I missed?) See for instance Foxby's "Isomorphisms between complexes with applications to the homological theory of modules".

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