Section 15.73 (0A5W): Derived hom

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. $\square$

Comments (2)

Comment #7935 by Karl Schwede on December 20, 2022 at 12:06

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".

Comment #8177 by Aise Johan de Jong on March 01, 2023 at 12:07

This section is just about constructing the maps. In future sections we prove some of these maps are isomorphisms. See for example: Sections 15.98 and 15.99.

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The Stacks project

15.73 Derived hom

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