The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

15.68 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

15.68.0.1
\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 15.67. A generalization of this construction is discussed in Differential Graded Algebra, Section 22.21. From (15.67.0.1) and Derived Categories, Lemma 13.29.2 that we have

15.68.0.2
\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 15.68.1. 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 (15.68.0.1) 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 15.67.1. 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 15.68.2. 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. $\square$

Lemma 15.68.3. 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 15.67.3. We omit the proof that this is functorial in all three objects of $D(R)$. $\square$

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. $\square$

Lemma 15.68.5. 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 15.67.5. $\square$

Lemma 15.68.6. 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 15.68.5 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 15.67.6. $\square$


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