Section 21.35 (08J7): Internal hom in the derived category

21.35 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

21.35.0.1

\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 21.34. This is well defined by Lemma 21.34.7. 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 (21.35.0.1) we compute the cohomology groups of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$.

Lemma 21.35.1. 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 21.34.8. Hence we can compute cohomology over $U$ by simply taking sections over $U$ and the result follows from Lemma 21.34.6. $\square$

Lemma 21.35.2. 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 (21.35.0.1) 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 21.34.1. 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 21.34.8) 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 21.35.1 we obtain (21.35.0.1). $\square$

Lemma 21.35.3. 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 21.20.1. $\square$

Lemma 21.35.4. 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 21.35.5. 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 21.34.5. 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 21.35.6. 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 21.34.2 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 21.35.7. 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 21.34.3. $\square$

Lemma 21.35.8. 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 21.34.4. $\square$

Lemma 21.35.9. 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

21.35.9.1

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

which induces a canonical map

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

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

Proof. The map (21.35.9.1) is a special case of Lemma 21.35.6 using the identification $M = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}, M)$. $\square$

Remark 21.35.10. 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 (21.35.0.1) 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 21.19.7) 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 21.35.6 (with $\mathcal{O}_\mathcal {C}$ in one of the spots).

Remark 21.35.11. 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 (21.35.0.1) proved in Lemma 21.35.2 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 21.18.4 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 (21.35.0.1).

Remark 21.35.12. 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 21.35.11.

Comments (5)

Comment #6370 by Hadi Hedayatzadeh on July 14, 2021 at 04:36

Typo: four lines after equation (21.34.0.1), "Then we set" is written twice.

Comment #7129 by Hao Peng on March 23, 2022 at 05:23

In 08JA, I guess the correct expression is $H^0(\Gamma(U, RHom(L, M)))$ rather than $H^0(U, RHom(L, M))$ .

Comment #7133 by Hao Peng on March 23, 2022 at 07:09

Just to point out it seems tags 0A97, 0A98, 0BYU, 0A99 follow from tag 08J9

Comment #7287 by Johan on May 03, 2022 at 14:29

@#7129: The notation $H^0(U, -)$ denotes the exact functor $D(\mathcal{O}) \to \textit{Ab}$ which you get by representing an object $K$ of $D(\mathcal{O})$ by a K-injective complex $\mathcal{I}^\bullet$ and then setting $H^0(U, K) = H^0(\mathcal{I}^\bullet(U))$ . Another way to say this would be that $H^0(U, K) = H^0(R\Gamma(U, K|_U)) = H^0(R\Gamma(U, K))$ . I have now explicitly introduced this notation in the section on unbounded cohomology, see these changes.

Comment #7291 by Johan on May 03, 2022 at 17:01

@#7133: Yes, I agree. But I think it is too much work to do the change and also I really want to know that the morphisms we get are the same (on underlying complexes of modules) as the ones we get from the low level constructions in the chapter "More on Algebra". So I have made the corresponding changes to that chapter and I hope that people who really want to get to the bottom of this will eventually get to it.

The Stacks project

21.35 Internal hom in the derived category

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