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*}

Lemma 7.25.4. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor $j_{U!}$ gives an equivalence of categories

\[ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \]

Proof. Let us denote objects of $\mathcal{C}/U$ as pairs $(X, a)$ where $X$ is an object of $\mathcal{C}$ and $a : X \to U$ is a morphism of $\mathcal{C}$. Similarly, objects of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ are pairs $(\mathcal{F}, \varphi )$. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ sends $\mathcal{G}$ to the pair $(j_{U!}\mathcal{G}, \gamma )$ where $\gamma $ is the composition of $j_{U!}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}* = h_ U^\# $.

Let us construct a functor from $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$. Suppose that $(\mathcal{F}, \varphi )$ is given. For an object $(X, a)$ of $\mathcal{C}/U$ we consider the set $\mathcal{F}_\varphi (X, a)$ of elements $s \in \mathcal{F}(X)$ which under $\varphi $ map to the image of $a \in \mathop{Mor}\nolimits _\mathcal {C}(X, U) = h_ U(X)$ in $h_ U^\# (X)$. It is easy to see that $(X, a) \mapsto \mathcal{F}_\varphi (X, a)$ is a sheaf on $\mathcal{C}/U$. Clearly, the rule $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ defines a functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U)$.

Consider also the functor $\textit{PSh}(\mathcal{C})/h_ U \to \textit{PSh}(\mathcal{C}/U)$, $(\mathcal{F}, \varphi ) \mapsto \mathcal{F}_\varphi $ where $\mathcal{F}_\varphi (X, a)$ is defined as the set of elements of $\mathcal{F}(X)$ mapping to $a \in h_ U(X)$. We claim that the diagram

\[ \xymatrix{ \textit{PSh}(\mathcal{C})/h_ U \ar[r] \ar[d] & \textit{PSh}(\mathcal{C}/U) \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) } \]

commutes, where the vertical arrows are given by sheafification. To see this1, it suffices to prove that the construction commutes with the functor $\mathcal{F} \mapsto \mathcal{F}^+$ of Lemmas 7.10.3 and 7.10.4 and Theorem 7.10.10. Commutation with $\mathcal{F} \mapsto \mathcal{F}^+$ follows from the fact that given $(X, a)$ the categories of coverings of $(X, a)$ in $\mathcal{C}/U$ and coverings of $X$ in $\mathcal{C}$ are canonically identified.

Next, let $\textit{PSh}(\mathcal{C}/U) \to \textit{PSh}(\mathcal{C})/h_ U$ send $\mathcal{G}$ to the pair $(j_{U!}^{PSh}\mathcal{G}, \gamma )$ where $j_{U!}^{PSh}\mathcal{G}$ the presheaf defined by the formula in Lemma 7.25.2 and $\gamma $ is the composition of $j_{U!}^{PSh}\mathcal{G} \to j_{U!}*$ with the identification $j_{U!}^{PSh}* = h_ U$ (obvious from the formula). Then it is immediately clear that the diagram

\[ \xymatrix{ \textit{PSh}(\mathcal{C}/U) \ar[r] \ar[d] & \textit{PSh}(\mathcal{C})/h_ U \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# } \]

commutes, where the vertical arrows are sheafification. Putting everything together it suffices to show there are functorial isomorphisms $(j_{U!}^{PSh}\mathcal{G})_\gamma = \mathcal{G}$ for $\mathcal{G}$ in $\textit{PSh}(\mathcal{C}/U)$ and $j_{U!}^{PSh}\mathcal{F}_\varphi = \mathcal{F}$ for $(\mathcal{F}, \varphi )$ in $\textit{PSh}(\mathcal{C})/h_ U$. The value of the presheaf $(j_{U!}^{PSh}\mathcal{G})_\gamma $ on $(X, a)$ is the fibre of the map

\[ \coprod \nolimits _{a' : X \to U} \mathcal{G}(X, a') \to \mathop{Mor}\nolimits _\mathcal {C}(X, U) \]

over $a$ which is $\mathcal{G}(X, a)$. This proves the first equality. The value of the presheaf $j_{U!}^{PSh}\mathcal{F}_\varphi $ is on $X$ is

\[ \coprod \nolimits _{a : X \to U} \mathcal{F}_\varphi (X, a) = \mathcal{F}(X) \]

because given a set map $S \to S'$ the set $S$ is the disjoint union of its fibres. $\square$

[1] An alternative is to describe $\mathcal{F}_\varphi $ by the cartesian diagram
\[ \vcenter { \xymatrix{ \mathcal{F}_\varphi \ar[r] \ar[d] & {*} \ar[d] \\ \mathcal{F}|_{\mathcal{C}/U} \ar[r] & h_ U|_{\mathcal{C}/U} } } \quad \text{for presheaves and}\quad \vcenter { \xymatrix{ \mathcal{F}_\varphi \ar[r] \ar[d] & {*} \ar[d] \\ \mathcal{F}|_{\mathcal{C}/U} \ar[r] & h_ U^\# |_{\mathcal{C}/U} } } \]
for sheaves and use that restriction to $\mathcal{C}/U$ commutes with sheafification.

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