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*}

20.36 Internal hom in the derived category

Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $L, M$ be objects of $D(\mathcal{O}_ X)$. We would like to construct an object $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)$ of $D(\mathcal{O}_ X)$ such that for every third object $K$ of $D(\mathcal{O}_ X)$ there exists a canonical bijection

20.36.0.1
\begin{equation} \label{cohomology-equation-internal-hom} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L, M) \end{equation}

Observe that this formula defines $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)$ up to unique isomorphism by the Yoneda lemma (Categories, Lemma 4.3.5).

To construct such an object, choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$ and any complex of $\mathcal{O}_ X$-modules $\mathcal{L}^\bullet $ representing $L$. Then we set

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet ) \]

where the right hand side is the complex of $\mathcal{O}_ X$-modules constructed in Section 20.35. This is well defined by Lemma 20.35.7. We get a functor

\[ D(\mathcal{O}_ X)^{opp} \times D(\mathcal{O}_ X) \longrightarrow D(\mathcal{O}_ X), \quad (K, L) \longmapsto R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \]

As a prelude to proving (20.36.0.1) we compute the cohomology groups of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$.

Lemma 20.36.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $L, M$ be objects of $D(\mathcal{O}_ X)$. For every open $U$ we have

\[ H^0(U, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U) \]

and in particular $H^0(X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(L, M)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}_ X$-modules representing $M$ and a K-flat complex $\mathcal{L}^\bullet $ representing $L$. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )$ is K-injective by Lemma 20.35.8. Hence we can compute cohomology over $U$ by simply taking sections over $U$ and the result follows from Lemma 20.35.6. $\square$

Lemma 20.36.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_ X)$. With the construction as described above there is a canonical isomorphism

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} L, M) \]

in $D(\mathcal{O}_ X)$ functorial in $K, L, M$ which recovers (20.36.0.1) by taking $H^0(X, -)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$ and a K-flat complex of $\mathcal{O}_ X$-modules $\mathcal{L}^\bullet $ representing $L$. Let $\mathcal{H}^\bullet $ be the complex described above. For any complex of $\mathcal{O}_ X$-modules $\mathcal{K}^\bullet $ we have

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{I}^\bullet )) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ( \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ), \mathcal{I}^\bullet ) \]

by Lemma 20.35.1. Note that the left hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M))$ (use Lemma 20.35.8) and that the right hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} L, M)$. This proves the displayed formula of the lemma. Taking global sections and using Lemma 20.36.1 we obtain (20.36.0.1). $\square$

Lemma 20.36.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_ X)$. The construction of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ commutes with restrictions to opens, i.e., for every open $U$ we have $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K|_ U, L|_ U) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)|_ U$.

Proof. This is clear from the construction and Lemma 20.30.1. $\square$

Lemma 20.36.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. The bifunctor $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (- , -)$ transforms distinguished triangles into distinguished triangles in both variables.

Proof. This follows from the observation that the assignment

\[ (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{M}^\bullet ) \]

transforms a termwise split short exact sequences of complexes in either variable into a termwise split short exact sequence. Details omitted. $\square$

Lemma 20.36.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L, M$ be objects of $D(\mathcal{O}_ X)$. There is a canonical morphism

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} K \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L), M) \]

in $D(\mathcal{O}_ X)$ functorial in $K, L, M$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$, a K-injective complex $\mathcal{J}^\bullet $ representing $L$, and a K-flat complex $\mathcal{K}^\bullet $ representing $K$. The map is defined using the map

\[ \text{Tot}(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet ) \otimes _{\mathcal{O}_ X} \mathcal{K}^\bullet ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet ), \mathcal{I}^\bullet ) \]

of Lemma 20.35.3. By our particular choice of complexes the left hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} K$ and the right hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L), M)$. We omit the proof that this is functorial in all three objects of $D(\mathcal{O}_ X)$. $\square$

Lemma 20.36.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given $K, L, M$ in $D(\mathcal{O}_ X)$ there is a canonical morphism

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, M) \]

in $D(\mathcal{O}_ X)$ functorial in $K, L, M$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$, a K-injective complex $\mathcal{J}^\bullet $ representing $L$, and any complex of $\mathcal{O}_ X$-modules $\mathcal{K}^\bullet $ representing $K$. By Lemma 20.35.2 there is a map of complexes

\[ \text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet ) \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet ) \right) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{I}^\bullet ) \]

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

\[ \text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet ) \otimes _{\mathcal{O}_ X} \mathcal{H}^\bullet \right) \longrightarrow \text{Tot}\left( \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{J}^\bullet , \mathcal{I}^\bullet ) \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{K}^\bullet , \mathcal{J}^\bullet ) \right) \]

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

Lemma 20.36.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given $K, L, M$ in $D(\mathcal{O}_ X)$ there is a canonical morphism

\[ K \otimes _{\mathcal{O}_ X}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

in $D(\mathcal{O}_ X)$ functorial in $K, L, M$.

Proof. Choose a K-flat complex $\mathcal{K}^\bullet $ representing $K$, and a K-injective complex $\mathcal{I}^\bullet $ representing $L$, and choose any complex of $\mathcal{O}_ X$-modules $\mathcal{M}^\bullet $ representing $M$. Choose a quasi-isomorphism $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet ) \to \mathcal{J}^\bullet $ where $\mathcal{J}^\bullet $ is K-injective. Then we use the map

\[ \text{Tot}\left( \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{I}^\bullet ) \right) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet )) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{J}^\bullet ) \]

where the first map is the map from Lemma 20.35.4. $\square$

Lemma 20.36.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given $K, L$ in $D(\mathcal{O}_ X)$ there is a canonical morphism

\[ K \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

in $D(\mathcal{O}_ X)$ functorial in both $K$ and $L$.

Proof. Choose a K-flat complex $\mathcal{K}^\bullet $ representing $K$ and any complex of $\mathcal{O}_ X$-modules $\mathcal{L}^\bullet $ representing $L$. Choose a K-injective complex $\mathcal{J}^\bullet $ and a quasi-isomorphism $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet ) \to \mathcal{J}^\bullet $. Then we use

\[ \mathcal{K}^\bullet \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{L}^\bullet )) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , \mathcal{J}^\bullet ) \]

where the first map comes from Lemma 20.35.5. $\square$

Lemma 20.36.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $L$ be an object of $D(\mathcal{O}_ X)$. Set $L^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O}_ X)$. For $M$ in $D(\mathcal{O}_ X)$ there is a canonical map

20.36.9.1
\begin{equation} \label{cohomology-equation-eval} L^\vee \otimes ^\mathbf {L}_{\mathcal{O}_ X} M \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \end{equation}

which induces a canonical map

\[ H^0(X, L^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(L, M) \]

functorial in $M$ in $D(\mathcal{O}_ X)$.

Proof. The map (20.36.9.1) is a special case of Lemma 20.36.6 using the identification $M = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, M)$. $\square$

Remark 20.36.10. Let $f : X \to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_ X)$. We claim there is a canonical map

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Rf_*L, Rf_*K) \]

Namely, by (20.36.0.1) this is the same thing as a map $Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \to Rf_*K$. For this we can use the composition

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \to Rf_*(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \to Rf_*K \]

where the first arrow is the relative cup product (Remark 20.29.6) and the second arrow is $Rf_*$ applied to the canonical map $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} L \to K$ coming from Lemma 20.36.6 (with $\mathcal{O}_ X$ in one of the spots).

Remark 20.36.11. Let $h : X \to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_ Y)$. We claim there is a canonical map

\[ Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lh^*K, Lh^*L) \]

in $D(\mathcal{O}_ X)$. Namely, by (20.36.0.1) proved in Lemma 20.36.2 such a map is the same thing as a map

\[ Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} Lh^*K \longrightarrow Lh^*L \]

The source of this arrow is $Lh^*(\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} K)$ by Lemma 20.28.3 hence it suffices to construct a canonical map

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \otimes ^\mathbf {L} K \longrightarrow L. \]

For this we take the arrow corresponding to

\[ \text{id} : R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \]

via (20.36.0.1).

Remark 20.36.12. Suppose that

\[ \xymatrix{ X' \ar[r]_ h \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

is a commutative diagram of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_ X)$. We claim there exists a canonical base change map

\[ Lg^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R(f')_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lh^*K, Lh^*L) \]

in $D(\mathcal{O}_{S'})$. Namely, we take the map adjoint to the composition

\begin{align*} L(f')^*Lg^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) & = Lh^*Lf^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \\ & \to Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \\ & \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lh^*K, Lh^*L) \end{align*}

where the first arrow uses the adjunction mapping $Lf^*Rf_* \to \text{id}$ and the second arrow is the canonical map constructed in Remark 20.36.11.


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