## 24.29 Derived pushforward

The existence of enough K-injective guarantees that we can take the right derived functor of any exact functor on the homotopy category.

Lemma 24.29.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Then any exact functor

$T : K(\textit{Mod}(\mathcal{A}, \text{d})) \longrightarrow \mathcal{D}$

of triangulated categories has a right derived extension $RT : D(\mathcal{A}, \text{d}) \to \mathcal{D}$ whose value on a graded injective and K-injective differential graded $\mathcal{A}$-module $\mathcal{I}$ is $T(\mathcal{I})$.

Proof. By Theorem 24.25.13 for any (right) differential graded $\mathcal{A}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{M} \to \mathcal{I}$ where $\mathcal{I}$ is a graded injective and K-injective differential graded $\mathcal{A}$-module. Hence by Derived Categories, Lemma 13.14.15 it suffices to show that given a quasi-isomorphism $\mathcal{I} \to \mathcal{I}'$ of differential graded $\mathcal{A}$-modules which are both graded injective and K-injective then $T(\mathcal{I}) \to T(\mathcal{I}')$ is an isomorphism. This is true because the map $\mathcal{I} \to \mathcal{I}'$ is an isomorphism in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ as follows for example from Lemma 24.26.7 (or one can deduce it from Lemma 24.25.10). $\square$

There are a number of functors we have already seen to which this applies. Here are two examples.

Definition 24.29.2. Derived internal hom and derived pushforward.

1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. The right derived extension

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}(\mathcal{N}, -) : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d})$

of the internal hom functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, -)$ is called derived internal hom.

2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal {D}$-algebras. The right derived extension

$Rf_* : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d})$

of the pushforward $f_*$ is called derived pushforward.

It turns out that $Rf_* : D(\mathcal{A}, \text{d}) \to D(\mathcal{B}, \text{d})$ agrees with derived pusforward on underlying complexes of $\mathcal{O}$-modules, see Lemma 24.29.8.

These functors are the adjoints of derived pullback and derived tensor product.

Lemma 24.29.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. Then

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}(\mathcal{N}, -) : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d})$

$- \otimes _\mathcal {A}^\mathbf {L} \mathcal{N} : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d})$

Proof. This follows from Derived Categories, Lemma 13.30.1 and Lemma 24.17.3. $\square$

Lemma 24.29.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal {D}$-algebras. Then

$Rf_* : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d})$

$Lf^* : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d})$

Proof. This follows from Derived Categories, Lemma 13.30.1 and Lemma 24.18.1. $\square$

Next, we discuss what happens in the situation considered in Section 24.28.

Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. Suppose we are given a map

$\varphi : f^{-1}\mathcal{B} \to \mathcal{A}$

of differential graded $f^{-1}\mathcal{O}_\mathcal {D}$-algebras. By the adjunction of restriction and extension of scalars, this is the same thing as a map $\varphi : f^*\mathcal{B} \to \mathcal{A}$ of differential graded $\mathcal{O}_\mathcal {C}$-algebras or equivalently $\varphi$ can be viewed as a map

$\varphi : \mathcal{B} \to f_*\mathcal{A}$

of differential graded $\mathcal{O}_\mathcal {D}$-algebras. See Remark 24.12.2.

In addition to the above, let $\mathcal{A}'$ be a second differential graded $\mathcal{O}_\mathcal {C}$-algebra and let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{A}')$-bimodule. In this setting we can consider the functor

$\textit{Mod}(\mathcal{A}', \text{d}) \longrightarrow \textit{Mod}(\mathcal{B}, \text{d}),\quad \mathcal{M} \longmapsto f_*\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_{\mathcal{A}'}(\mathcal{N}, \mathcal{M})$

Observe that this extends to a functor

$\textit{Mod}^{dg}(\mathcal{A}', \text{d}) \longrightarrow \textit{Mod}^{dg}(\mathcal{B}, \text{d}),\quad \mathcal{M} \longmapsto f_*\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_{\mathcal{A}'}(\mathcal{N}, \mathcal{M})$

of differential graded categories by the discussion in Sections 24.18 and 24.17. It follows formally that we also obtain an exact functor

24.29.4.1
$$\label{sdga-equation-pushforward} K(\textit{Mod}(\mathcal{A}', \text{d})) \longrightarrow K(\textit{Mod}(\mathcal{B}, \text{d})),\quad \mathcal{M} \longmapsto f_*\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_{\mathcal{A}'}(\mathcal{N}, \mathcal{M})$$

of triangulated categories.

Lemma 24.29.5. In the situation above, denote $RT : D(\mathcal{A}', \text{d}) \to D(\mathcal{B}, \text{d})$ the right derived extension of (24.29.4.1). Then we have

$RT(\mathcal{M}) = Rf_* R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{N}, \mathcal{M})$

functorially in $\mathcal{M}$.

Proof. By Lemmas 24.17.3 and 24.18.1 the functor (24.29.4.1) is right adjoint to the functor (24.28.0.1). By Derived Categories, Lemma 13.30.1 the functor $RT$ is right adjoint to the functor of Lemma 24.28.1 which is equal to $Lf^*(-) \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$ by Lemma 24.28.3. By Lemmas 24.29.3 and 24.29.4 the functor $Lf^*(-) \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$ is left adjoint to $Rf_* R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{N}, -)$ Thus we conclude by uniqueness of adjoints. $\square$

Lemma 24.29.6. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ and $(g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}''), \mathcal{O}'')$ be morphisms of ringed topoi. Let $\mathcal{A}$, $\mathcal{A}'$, and $\mathcal{A}''$ be a differential graded $\mathcal{O}$-algebra, $\mathcal{O}'$-algebra, and $\mathcal{O}''$-algebra. Let $\varphi : \mathcal{A}' \to f_*\mathcal{A}$ and $\varphi ' : \mathcal{A}'' \to g_*\mathcal{A}'$ be a homomorphism of differential graded $\mathcal{O}'$-algebras and $\mathcal{O}''$-algebras. Then we have $R(g \circ f)_* = Rg_* \circ Rf_* : D(\mathcal{A}, \text{d}) \to D(\mathcal{A}'', \text{d})$.

Proof. Follows from Lemmas 24.28.4 and 24.29.4 and uniqueness of adjoints. $\square$

Lemma 24.29.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{A}'$, $\mathcal{A}''$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ and $\mathcal{N}'$ be a differential graded $(\mathcal{A}, \mathcal{A}')$-bimodule and $(\mathcal{A}', \mathcal{A}'')$-bimodule. Assume that the canonical map

$\mathcal{N} \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' \longrightarrow \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}'$

in $D(\mathcal{A}'', \text{d})$ is a quasi-isomorphism. Then we have

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{A}''} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}', -) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{A}'}(\mathcal{N}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{A}''}(\mathcal{N}', -))$

as functors $D(\mathcal{A}'', \text{d}) \to D(\mathcal{A}, \text{d})$.

Proof. Follows from Lemmas 24.28.7 and 24.29.3 and uniqueness of adjoints. $\square$

Lemma 24.29.8. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal {D}$-algebras. The diagram

$\xymatrix{ D(\mathcal{A}, \text{d}) \ar[d]_{Rf_*} \ar[rr]_{forget} & & D(\mathcal{O}_\mathcal {C}) \ar[d]^{Rf_*} \\ D(\mathcal{B}, \text{d}) \ar[rr]^{forget} & & D(\mathcal{O}_\mathcal {D}) }$

commutes.

Proof. Besides identifying some categories, this lemma follows immediately from Lemma 24.29.6.

We may view $(\mathcal{O}_\mathcal {C}, 0)$ as a differential graded $\mathcal{O}_\mathcal {C}$-algebra by placing $\mathcal{O}_\mathcal {C}$ in degree $0$ and endowing it with the zero differential. It is clear that we have

$\textit{Mod}(\mathcal{O}_\mathcal {C}, 0) = \text{Comp}(\mathcal{O}_\mathcal {C}) \quad \text{and}\quad D(\mathcal{O}_\mathcal {C}, 0) = D(\mathcal{O}_\mathcal {C})$

Via this identification the forgetful functor $\textit{Mod}(\mathcal{A}, \text{d}) \to \text{Comp}(\mathcal{O}_\mathcal {C})$ is the “pushforward” $\text{id}_{\mathcal{C}, *}$ defined in Section 24.18 corresponding to the identity morphism $\text{id}_\mathcal {C} : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{C}, \mathcal{O}_\mathcal {C})$ of ringed topoi and the map $(\mathcal{O}_\mathcal {C}, 0) \to (\mathcal{A}, \text{d})$ of differential graded $\mathcal{O}_\mathcal {C}$-algebras. Since $\text{id}_{\mathcal{C}, *}$ is exact, we immediately see that

$R\text{id}_{\mathcal{C}, *} = forget : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{O}_\mathcal {C}, 0) = D(\mathcal{O}_\mathcal {C})$

The exact same reasoning shows that

$R\text{id}_{\mathcal{D}, *} = forget : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{O}_\mathcal {D}, 0) = D(\mathcal{O}_\mathcal {D})$

Moreover, the construction of $Rf_* : D(\mathcal{O}_\mathcal {C}) \to D(\mathcal{O}_\mathcal {D})$ of Cohomology on Sites, Section 21.19 agrees with the construction of $Rf_* : D(\mathcal{O}_\mathcal {C}, 0) \to D(\mathcal{O}_\mathcal {D}, 0)$ in Definition 24.29.2 as both functors are defined as the right derived extension of pushforward on underlying complexes of modules. By Lemma 24.29.6 we see that both $Rf_* \circ R\text{id}_{\mathcal{C}, *}$ and $R\text{id}_{\mathcal{D}, *} \circ Rf_*$ are the derived functors of $f_* \circ forget = forget \circ f_*$ and hence equal by uniqueness of adjoints. $\square$

Lemma 24.29.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a differential graded $\mathcal{O}$-algebra. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. Let $n \in \mathbf{Z}$. We have

$H^ n(\mathcal{C}, \mathcal{M}) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{A}, \mathcal{M}[n])$

where on the left hand side we have the cohomology of $\mathcal{M}$ viewed as a complex of $\mathcal{O}$-modules.

Proof. To prove the formula, observe that

$R\Gamma (\mathcal{C}, \mathcal{M}) = \Gamma (\mathcal{C}, \mathcal{I})$

where $\mathcal{M} \to \mathcal{I}$ is a quasi-isomorphism to a graded injective and K-injective differential graded $\mathcal{A}$-module $\mathcal{I}$ (combine Lemmas 24.29.1 and 24.29.8). By Lemma 24.26.7 we have

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{A}, \mathcal{M}[n]) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I}[n]) = H^0(\Gamma (\mathcal{C}, \mathcal{I}[n])) = H^ n(\Gamma (\mathcal{C}, \mathcal{I}))$

Combining these two results we obtain our equality. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).