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 7.41.2. Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. Assume that for any object $U$ of $\mathcal{C}$ and any covering $\{ V_ j \to u(U)\} $ in $\mathcal{D}$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ such that the map of sheaves

\[ \coprod h_{u(U_ i)}^\# \to h_{u(U)}^\# \]

factors through the map of sheaves

\[ \coprod h_{V_ j}^\# \to h_{u(U)}^\# . \]

Then $f_*$ transforms surjective maps of sheaves into surjective maps of sheaves.

Proof. Let $a : \mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves on $\mathcal{D}$. Let $U$ be an object of $\mathcal{C}$ and let $s \in f_*\mathcal{G}(U) = \mathcal{G}(u(U))$. By assumption there exists a covering $\{ V_ j \to u(U)\} $ and sections $s_ j \in \mathcal{F}(V_ j)$ with $a(s_ j) = s|_{V_ j}$. Now we may think of the sections $s$, $s_ j$ and $a$ as giving a commutative diagram of maps of sheaves

\[ \xymatrix{ \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ h_{u(U)}^\# \ar[r]^ s & \mathcal{G} } \]

By assumption there exists a covering $\{ U_ i \to U\} $ such that we can enlarge the commutative diagram above as follows

\[ \xymatrix{ & \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ \coprod h_{u(U_ i)}^\# \ar[r] \ar[ur] & h_{u(U)}^\# \ar[r]^ s & \mathcal{G} } \]

Because $\mathcal{F}$ is a sheaf the map from the left lower corner to the right upper corner corresponds to a family of sections $s_ i \in \mathcal{F}(u(U_ i))$, i.e., sections $s_ i \in f_*\mathcal{F}(U_ i)$. The commutativity of the diagram implies that $a(s_ i)$ is equal to the restriction of $s$ to $U_ i$. In other words we have shown that $f_*a$ is a surjective map of sheaves. $\square$


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