Lemma 21.38.5. Assumption and notation as in Situation 21.38.3.

1. There are left adjoints $g_! : \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \to \textit{Mod}(\mathcal{O}_\mathcal {C})$ and $g_!^{\textit{Ab}} : \textit{Ab}(\mathcal{C}') \to \textit{Ab}(\mathcal{C})$ to $g^* = g^{-1}$ on modules and on abelian sheaves.

2. The diagram

$\xymatrix{ \textit{Mod}(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{g_!} & \textit{Mod}(\mathcal{O}_\mathcal {C}) \ar[d] \\ \textit{Ab}(\mathcal{C}') \ar[r]^{g_!^{\textit{Ab}}} & \textit{Ab}(\mathcal{C}) }$

commutes.

3. There are left adjoints $Lg_! : D(\mathcal{O}_{\mathcal{C}'}) \to D(\mathcal{O}_\mathcal {C})$ and $Lg_!^{\textit{Ab}} : D(\mathcal{C}') \to D(\mathcal{C})$ to $g^* = g^{-1}$ on derived categories of modules and abelian sheaves.

4. The diagram

$\xymatrix{ D(\mathcal{O}_{\mathcal{C}'}) \ar[d] \ar[r]_{Lg_!} & D(\mathcal{O}_\mathcal {C}) \ar[d] \\ D(\mathcal{C}') \ar[r]^{Lg_!^{\textit{Ab}}} & D(\mathcal{C}) }$

commutes.

Proof. The functor $u$ is continuous and cocontinuous Stacks, Lemma 8.10.3. Hence the existence of the functors $g_!$, $g_!^{\textit{Ab}}$, $Lg_!$, and $Lg_!^{\textit{Ab}}$ can be found in Modules on Sites, Sections 18.16 and 18.41 and Section 21.37.

To prove (2) it suffices to show that the canonical map

$g_!^{\textit{Ab}}j_{U'!}\mathcal{O}_{U'} \to j_{u(U')!}\mathcal{O}_{u(U')}$

is an isomorphism for all objects $U'$ of $\mathcal{C}'$, see Modules on Sites, Remark 18.41.2. Similarly, to prove (4) it suffices to show that the canonical map

$Lg_!^{\textit{Ab}}j_{U'!}\mathcal{O}_{U'} \to j_{u(U')!}\mathcal{O}_{u(U')}$

is an isomorphism in $D(\mathcal{C})$ for all objects $U'$ of $\mathcal{C}'$, see Remark 21.37.3. This will also imply the previous formula hence this is what we will show.

We will use that for a localization morphism $j$ the functors $j_!$ and $j_!^{\textit{Ab}}$ agree (see Modules on Sites, Remark 18.19.6) and that $j_!$ is exact (Modules on Sites, Lemma 18.19.3). Let us adopt the notation of Lemma 21.38.4. Since $Lg_!^{\textit{Ab}} \circ j_{U'!} = j_{U!} \circ L(g')^{\textit{Ab}}_!$ (by commutativity of Sites, Lemma 7.28.4 and uniqueness of adjoint functors) it suffices to prove that $L(g')^{\textit{Ab}}_!\mathcal{O}_{U'} = \mathcal{O}_ U$. Using the results of Lemma 21.38.4 we have for any object $E$ of $D(\mathcal{C}/u(U'))$ the following sequence of equalities

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}/U)}(L(g')_!^{\textit{Ab}}\mathcal{O}_{U'}, E) & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}'/U')}(\mathcal{O}_{U'}, (g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}'/U')}((\pi '_{U'})^{-1}\mathcal{O}_ V, (g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, R\pi '_{U', *}(g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, (\sigma ')^{-1}(g')^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, \sigma ^{-1}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{D}/V)}(\mathcal{O}_ V, \pi _{U, *}E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}/U)}(\pi _ U^{-1}\mathcal{O}_ V, E) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{C}/U)}(\mathcal{O}_ U, E) \end{align*}

By Yoneda's lemma we conclude. $\square$

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