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*}

7.24 Existence of lower shriek

In this section we discuss some cases of morphisms of topoi $f$ for which $f^{-1}$ has a left adjoint $f_!$.

Lemma 7.24.1. Let $\mathcal{C}$, $\mathcal{D}$ be two sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Let $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ be a subset such that

  1. for $V \in E$ there exists a sheaf $\mathcal{G}$ on $\mathcal{C}$ such that $f^{-1}\mathcal{F}(V) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, \mathcal{F})$ functorially for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$,

  2. every object of $\mathcal{D}$ has a covering by objects of $E$.

Then $f^{-1}$ has a left adjoint $f_!$.

Proof. By the Yoneda lemma (Categories, Lemma 4.3.5) the sheaf $\mathcal{G}_ V$ corresponding to $V \in E$ is defined up to unique isomorphism by the formula $f^{-1}\mathcal{F}(V) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}_ V, \mathcal{F})$. Recall that $f^{-1}\mathcal{F}(V) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(h_ V^\# , f^{-1}\mathcal{F})$. Denote $i_ V : h_ V^\# \to f^{-1}\mathcal{G}_ V$ the map corresponding to $\text{id}$ in $\mathop{Mor}\nolimits (\mathcal{G}_ V, \mathcal{G}_ V)$. Functoriality in (1) implies that the bijection is given by

\[ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}_ V, \mathcal{F}) \to \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(h_ V^\# , f^{-1}\mathcal{F}),\quad \varphi \mapsto f^{-1}\varphi \circ i_ V \]

For any $V_1, V_2 \in E$ there is a canonical map

\[ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(h^\# _{V_2}, h^\# _{V_1}) \to \mathop{\mathrm{Hom}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}_{V_2}, \mathcal{G}_{V_1}),\quad \varphi \mapsto f_!(\varphi ) \]

which is characterized by $f^{-1}(f_!(\varphi )) \circ i_{V_2} = i_{V_1} \circ \varphi $. Note that $\varphi \mapsto f_!(\varphi )$ is compatible with composition; this can be seen directly from the characterization. Hence $h_ V^\# \mapsto \mathcal{G}_ V$ and $\varphi \mapsto f_!\varphi $ is a functor from the full subcategory of $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ whose objects are the $h_ V^\# $.

Let $J$ be a set and let $J \to E$, $j \mapsto V_ j$ be a map. Then we have a functorial bijection

\[ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\coprod \mathcal{G}_{V_ j}, \mathcal{F}) \longrightarrow \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(\coprod h_{V_ j}^\# , f^{-1}\mathcal{F}) \]

using the product of the bijections above. Hence we can extend the functor $f_!$ to the full subcategory of $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ whose objects are coproducts of $h_ V^\# $ with $V \in E$.

Given an arbitrary sheaf $\mathcal{H}$ on $\mathcal{D}$ we choose a coequalizer diagram

\[ \xymatrix{ \mathcal{H}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{H}_0 \ar[r] & \mathcal{H} } \]

where $\mathcal{H}_ i = \coprod h_{V_{i, j}}^\# $ is a coproduct with $V_{i, j} \in E$. This is possible by assumption (2), see Lemma 7.12.5 (for those worried about set theoretical issues, note that the construction given in Lemma 7.12.5 is canonical). Define $f_!(\mathcal{H})$ to be the sheaf on $\mathcal{C}$ which makes

\[ \xymatrix{ f_!\mathcal{H}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & f_!\mathcal{H}_0 \ar[r] & f_!\mathcal{H} } \]

a coequalizer diagram. Then

\begin{align*} \mathop{Mor}\nolimits (f_!\mathcal{H}, \mathcal{F}) & = \text{Equalizer}( \xymatrix{ \mathop{Mor}\nolimits (f_!\mathcal{H}_0, \mathcal{F}) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{Mor}\nolimits (f_!\mathcal{H}_1, \mathcal{F}) } ) \\ & = \text{Equalizer}( \xymatrix{ \mathop{Mor}\nolimits (\mathcal{H}_0, f^{-1}\mathcal{F}) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{Mor}\nolimits (\mathcal{H}_1, f^{-1}\mathcal{F}) } ) \\ & = \mathop{\mathrm{Hom}}\nolimits (\mathcal{H}, f^{-1}\mathcal{F}) \end{align*}

Hence we see that we can extend $f_!$ to the whole category of sheaves on $\mathcal{D}$. $\square$


Comments (3)

Comment #3071 by Dario Weißmann on

After the diagram used to define the sentence should be finished with 'exact' or 'an equalizer diagram'.

Comment #3073 by Dario Weißmann on

*coequalizer

Also, a few lines above 'an coequalizer' should be 'a coequalizer'


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