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*}

18.40 Lower shriek for modules

In this section we extend the construction of $g_!$ discussed in Section 18.16 to the case of sheaves of modules.

Lemma 18.40.1. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous and cocontinuous functor between sites. Denote $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ the associated morphism of topoi. Let $\mathcal{O}_\mathcal {D}$ be a sheaf of rings on $\mathcal{D}$. Set $\mathcal{O}_\mathcal {C} = g^{-1}\mathcal{O}_\mathcal {D}$. Hence $g$ becomes a morphism of ringed topoi with $g^* = g^{-1}$. In this case there exists a functor

\[ g_! : \textit{Mod}(\mathcal{O}_\mathcal {C}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal {D}) \]

which is left adjoint to $g^*$.

Proof. Let $U$ be an object of $\mathcal{C}$. For any $\mathcal{O}_\mathcal {D}$-module $\mathcal{G}$ we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {C}}(j_{U!}\mathcal{O}_ U, g^{-1}\mathcal{G}) & = g^{-1}\mathcal{G}(U) \\ & = \mathcal{G}(u(U)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(j_{u(U)!}\mathcal{O}_{u(U)}, \mathcal{G}) \end{align*}

because $g^{-1}$ is described by restriction, see Sites, Lemma 7.21.5. Of course a similar formula holds a direct sum of modules of the form $j_{U!}\mathcal{O}_ U$. By Homology, Lemma 12.26.6 and Lemma 18.28.7 we see that $g_!$ exists. $\square$

Remark 18.40.2. Warning! Let $u : \mathcal{C} \to \mathcal{D}$, $g$, $\mathcal{O}_\mathcal {D}$, and $\mathcal{O}_\mathcal {C}$ be as in Lemma 18.40.1. In general it is not the case that the diagram

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

commutes (here $g^{Ab}_!$ is the one from Lemma 18.16.2). There is a transformation of functors

\[ g_!^{Ab} \circ forget \longrightarrow forget \circ g_! \]

From the proof of Lemma 18.40.1 we see that this is an isomorphism if and only if $g^{Ab}_!j_{U!}\mathcal{O}_ U \to g_!j_{U!}\mathcal{O}_ U$ is an isomorphism for all objects $U$ of $\mathcal{C}$. Since we have $g_!j_{U!}\mathcal{O}_ U = j_{u(U)!}\mathcal{O}_{u(U)}$ this holds if and only if

\[ g^{Ab}_!j_{U!}\mathcal{O}_ U \longrightarrow j_{u(U)!}\mathcal{O}_{u(U)} \]

is an isomorphism for all objects $U$ of $\mathcal{C}$. Note that for such a $U$ we obtain a commutative diagram

\[ \xymatrix{ \mathcal{C}/U \ar[r]_-{u'} \ar[d]_{j_ U} & \mathcal{D}/u(U) \ar[d]^{j_{u(U)}} \\ \mathcal{C} \ar[r]^ u & \mathcal{D} } \]

of cocontinuous functors of sites, see Sites, Lemma 7.28.4 and therefore $g^{Ab}_!j_{U!} = j_{u(U)!}(g')^{Ab}_!$ where $g' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/u(U))$ is the morphism of topoi induced by the cocontinuous functor $u'$. Hence we see that $g_! = g^{Ab}_!$ if the canonical map
\begin{equation} \label{sites-modules-equation-compare-on-localizations} (g')^{Ab}_!\mathcal{O}_ U \longrightarrow \mathcal{O}_{u(U)} \end{equation}

is an isomorphism for all objects $U$ of $\mathcal{C}$.

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