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*}

21.34 Internal hom in the derived category

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

21.34.0.1
\begin{equation} \label{sites-cohomology-equation-internal-hom} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K \otimes _\mathcal {O}^\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 of $\mathcal{O}$-modules $\mathcal{I}^\bullet $ representing $M$ and any complex of $\mathcal{O}$-modules $\mathcal{L}^\bullet $ representing $L$. Then we set 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 $\mathcal{O}$-modules constructed in Section 21.33. This is well defined by Lemma 21.33.7. We get a functor

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

As a prelude to proving (21.34.0.1) we compute the cohomology groups of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$.

Lemma 21.34.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. For every object $U$ of $\mathcal{C}$ 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 we have $H^0(\mathcal{C}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(L, M)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}$-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 21.33.8. Hence we can compute cohomology over $U$ by simply taking sections over $U$ and the result follows from Lemma 21.33.6. $\square$

Lemma 21.34.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of $D(\mathcal{O})$. With the construction as described above there is a canonical isomorphism

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

in $D(\mathcal{O})$ functorial in $K, L, M$ which recovers (21.34.0.1) on taking $H^0(\mathcal{C}, -)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$ and a K-flat complex of $\mathcal{O}$-modules $\mathcal{L}^\bullet $ representing $L$. For any complex of $\mathcal{O}$-modules $\mathcal{K}^\bullet $ we have

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

by Lemma 21.33.1. 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 21.33.8) and that the right hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K \otimes _\mathcal {O}^\mathbf {L} L, M)$. This proves the displayed formula of the lemma. Taking global sections and using Lemma 21.34.1 we obtain (21.34.0.1). $\square$

Lemma 21.34.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. The construction of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ commutes with restrictions, i.e., for every object $U$ of $\mathcal{C}$ 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 21.21.1. $\square$

Lemma 21.34.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. 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 21.34.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L, M$ be objects of $D(\mathcal{O})$. There is a canonical morphism

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

in $D(\mathcal{O})$ 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

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

of Lemma 21.33.3. By our particular choice of complexes the left hand side represents $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \otimes _\mathcal {O}^\mathbf {L} K$ 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 $D(\mathcal{O})$. $\square$

slogan

Lemma 21.34.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\mathcal{O})$ there is a canonical morphism

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

in $D(\mathcal{O})$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $M$, a K-injective complex $\mathcal{J}^\bullet $ representing $L$, and any complex of $\mathcal{O}$-modules $\mathcal{K}^\bullet $ representing $K$. By Lemma 21.33.2 there is a map of complexes

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

The complexes of $\mathcal{O}$-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

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

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

Lemma 21.34.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L, M$ in $D(\mathcal{O})$ there is a canonical morphism

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

in $D(\mathcal{O})$ 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 $\mathcal{O}$-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

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

where the first map is the map from Lemma 21.33.4. $\square$

Lemma 21.34.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L$ in $D(\mathcal{O})$ there is a canonical morphism

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

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

Proof. Choose a K-flat complex $\mathcal{K}^\bullet $ representing $K$ and any complex of $\mathcal{O}$-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} \mathcal{L}^\bullet ) \to \mathcal{J}^\bullet $. Then we use

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

where the first map comes from Lemma 21.33.5. $\square$

Lemma 21.34.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $L$ be an object of $D(\mathcal{O})$. Set $L^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, \mathcal{O})$. For $M$ in $D(\mathcal{O})$ there is a canonical map

21.34.9.1
\begin{equation} \label{sites-cohomology-equation-eval} L^\vee \otimes ^\mathbf {L}_\mathcal {O} M \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, M) \end{equation}

which induces a canonical map

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

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

Proof. The map (21.34.9.1) is a special case of Lemma 21.34.6 using the identification $M = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}, M)$. $\square$

Remark 21.34.10. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}_\mathcal {C})$. 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 (21.34.0.1) this is the same thing as a map $Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_\mathcal {D}}^\mathbf {L} Rf_*L \to Rf_*K$. For this we can use the composition

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

where the first arrow is the relative cup product (Remark 21.20.6) and the second arrow is $Rf_*$ applied to the canonical map $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} L \to K$ coming from Lemma 21.34.6 (with $\mathcal{O}_\mathcal {C}$ in one of the spots).

Remark 21.34.11. Let $h : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}')$. 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 $D(\mathcal{O})$. Namely, by (21.34.0.1) proved in Lemma 21.34.2 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 21.19.4 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 (21.34.0.1).

Remark 21.34.12. Suppose that

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_ h \ar[d]_{f'} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[d]^ f \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^ g & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) } \]

is a commutative diagram of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}_\mathcal {C})$. 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 $D(\mathcal{O}_{\mathcal{D}'})$. 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 21.34.11.


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