Lemma 21.36.2. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor of sites. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the corresponding morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings and set $\mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}$. The functor $g_! : \textit{Mod}(\mathcal{O}_\mathcal {C}) \to \textit{Mod}(\mathcal{O}_\mathcal {D})$ (see Modules on Sites, Lemma 18.40.1) has a left derived functor

$Lg_! : D(\mathcal{O}_\mathcal {C}) \longrightarrow D(\mathcal{O}_\mathcal {D})$

which is left adjoint to $g^*$. Moreover, for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have

$Lg_!(j_{U!}\mathcal{O}_ U) = g_!j_{U!}\mathcal{O}_ U = j_{u(U)!} \mathcal{O}_{u(U)}.$

where $j_{U!}$ and $j_{u(U)!}$ are extension by zero associated to the localization morphism $j_ U : \mathcal{C}/U \to \mathcal{C}$ and $j_{u(U)} : \mathcal{D}/u(U) \to \mathcal{D}$.

Proof. We are going to use Derived Categories, Proposition 13.28.2 to construct $Lg_!$. To do this we have to verify assumptions (1), (2), (3), (4), and (5) of that proposition. First, since $g_!$ is a left adjoint we see that it is right exact and commutes with all colimits, so (5) holds. Conditions (3) and (4) hold because the category of modules on a ringed site is a Grothendieck abelian category. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}_\mathcal {C}))$ be the collection of $\mathcal{O}_\mathcal {C}$-modules which are direct sums of modules of the form $j_{U!}\mathcal{O}_ U$. Note that $g_!j_{U!}\mathcal{O}_ U = j_{u(U)!} \mathcal{O}_{u(U)}$, see proof of Modules on Sites, Lemma 18.40.1. Every $\mathcal{O}_\mathcal {C}$-module is a quotient of an object of $\mathcal{P}$, see Modules on Sites, Lemma 18.28.7. Thus (1) holds. Finally, we have to prove (2). Let $\mathcal{K}^\bullet$ be a bounded above acyclic complex of $\mathcal{O}_\mathcal {C}$-modules with $\mathcal{K}^ n \in \mathcal{P}$ for all $n$. We have to show that $g_!\mathcal{K}^\bullet$ is exact. To do this it suffices to show, for every injective $\mathcal{O}_\mathcal {D}$-module $\mathcal{I}$ that

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {D})}( g_!\mathcal{K}^\bullet , \mathcal{I}[n]) = 0$

for all $n \in \mathbf{Z}$. Since $\mathcal{I}$ is injective we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {D})}( g_!\mathcal{K}^\bullet , \mathcal{I}[n]) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_\mathcal {D})}( g_!\mathcal{K}^\bullet , \mathcal{I}[n]) \\ & = H^ n(\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}( g_!\mathcal{K}^\bullet , \mathcal{I})) \\ & = H^ n(\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I})) \end{align*}

the last equality by the adjointness of $g_!$ and $g^{-1}$.

The vanishing of this group would be clear if $g^{-1}\mathcal{I}$ were an injective $\mathcal{O}_\mathcal {C}$-module. But $g^{-1}\mathcal{I}$ isn't necessarily an injective $\mathcal{O}_\mathcal {C}$-module as $g_!$ isn't exact in general. We do know that

$\mathop{\mathrm{Ext}}\nolimits ^ p_{\mathcal{O}_\mathcal {C}}( j_{U!}\mathcal{O}_ U, g^{-1}\mathcal{I}) = H^ p(U, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1$

Here the first equality follows from $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(j_{U!}\mathcal{O}_ U, \mathcal{H}) = \mathcal{H}(U)$ and taking derived functors and the vanishing of $H^ p(U, g^{-1}\mathcal{I})$ for $p >0$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ follows from Lemma 21.36.1. Since each $\mathcal{K}^{-q}$ is a direct sum of modules of the form $j_{U!}\mathcal{O}_ U$ we see that

$\mathop{\mathrm{Ext}}\nolimits ^ p_{\mathcal{O}_\mathcal {C}}(\mathcal{K}^{-q}, g^{-1}\mathcal{I}) = 0 \text{ for }p \geq 1\text{ and all }q$

Let us use the spectral sequence (see Example 21.32.1)

$E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits ^ p_{\mathcal{O}_\mathcal {C}}( \mathcal{K}^{-q}, g^{-1}\mathcal{I}) \Rightarrow \mathop{\mathrm{Ext}}\nolimits ^{p + q}_{\mathcal{O}_\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I}) = 0.$

Note that the spectral sequence abuts to zero as $\mathcal{K}^\bullet$ is acyclic (hence vanishes in the derived category, hence produces vanishing ext groups). By the vanishing of higher exts proved above the only nonzero terms on the $E_1$ page are the terms $E_1^{0, q} = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( \mathcal{K}^{-q}, g^{-1}\mathcal{I})$. We conclude that the complex $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I})$ is acyclic as desired.

Thus the left derived functor $Lg_!$ exists. It is left adjoint to $g^{-1} = g^* = Rg^* = Lg^*$, i.e., we have

21.36.2.1
$$\label{sites-cohomology-equation-to-prove} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {C})}(K, g^*L) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {D})}(Lg_!K, L)$$

by Derived Categories, Lemma 13.28.5. This finishes the proof. $\square$

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