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*}

7.45 Pullback maps

It sometimes happens that a site $\mathcal{C}$ does not have a final object. In this case we define the global section functor as follows.

Definition 7.45.1. The global sections of a presheaf of sets $\mathcal{F}$ over a site $\mathcal{C}$ is the set

\[ \Gamma (\mathcal{C}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(*, \mathcal{F}) \]

where $*$ is the final object in the category of presheaves on $\mathcal{C}$, i.e., the presheaf which associates to every object a singleton.

Of course the same definition applies to sheaves as well. Here is one way to compute global sections.

Lemma 7.45.2. Let $\mathcal{C}$ be a site. Let $a, b : V \to U$ be objects of $\mathcal{C}$ such that

\[ \xymatrix{ h_ V^\# \ar@<1ex>[r] \ar@<-1ex>[r] & h_ U^\# \ar[r] & {*} } \]

is a coequalizer in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. Then $\Gamma (\mathcal{C}, \mathcal{F})$ is the equalizer of $a^*, b^* : \mathcal{F}(U) \to \mathcal{F}(V)$.

Proof. Since $\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(h_ U^\# , \mathcal{F}) = \mathcal{F}(U)$ this is clear from the definitions. $\square$

Now, let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Then for any sheaf $\mathcal{F}$ on $\mathcal{C}$ there is a pullback map

\[ f^{-1} : \Gamma (\mathcal{C}, \mathcal{F}) \longrightarrow \Gamma (\mathcal{D}, f^{-1}\mathcal{F}) \]

Namely, as $f^{-1}$ is exact it transforms $*$ into $*$. We can generalize this a bit by considering a pair of sheaves $\mathcal{F}, \mathcal{G}$ on $\mathcal{C}, \mathcal{D}$ together with a map $f^{-1}\mathcal{F} \to \mathcal{G}$. Then we compose to get a map

\[ \Gamma (\mathcal{C}, \mathcal{F}) \longrightarrow \Gamma (\mathcal{D}, \mathcal{G}) \]

A slightly more general construction which occurs frequently in nature is the following. Suppose that we have a commutative diagram of morphisms of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar[rd]_ h \ar[rr]_ f & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[ld]^ g \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{B}) } \]

Next, suppose that we have a sheaf $\mathcal{F}$ on $\mathcal{C}$. Then there is a pullback map

\[ f^{-1} : g_*\mathcal{F} \longrightarrow h_*f^{-1}\mathcal{F} \]

Namely, it is just the map coming from the identification $h_*f^{-1}\mathcal{F} = g_*f_*f^{-1}\mathcal{F}$ together with the canonical map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ pushed down to $\mathcal{B}$. Again, if we have a pair of sheaves $\mathcal{F}, \mathcal{G}$ on $\mathcal{C}, \mathcal{D}$ together with a map $f^{-1}\mathcal{F} \to \mathcal{G}$, then we compose to get a map

\[ g_*\mathcal{F} \longrightarrow h_*\mathcal{G} \]

Restricting to sections over an object of $\mathcal{B}$ one recovers the pullback map on global sections in many cases, see (insert future reference here). A seemingly more general situation is where we have a commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \ar[d]_ h \ar[r]_ f & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ g \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{B}) \ar[r]^ e & \mathop{\mathit{Sh}}\nolimits (\mathcal{A}) } \]

and a sheaf $\mathcal{G}$ on $\mathcal{C}$. Then there is a map $e^{-1}g_*\mathcal{G} \to h_*f^{-1}\mathcal{G}$. Namely, this map is adjoint to a map $g_*\mathcal{G} \to e_*h_*f^{-1}\mathcal{G} = (e \circ h)_*f^{-1}\mathcal{G}$ which is the pullback map just described.


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