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 8.4.8. Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be stacks over $\mathcal{C}$. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$ be a $1$-morphism which is fully faithful. Then the following are equivalent

  1. $F$ is an equivalence,

  2. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for every $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{2, U})$ there exists a covering $\{ f_ i : U_ i \to U\} $ such that $f_ i^*x$ is in the essential image of the functor $F : \mathcal{S}_{1, U_ i} \to \mathcal{S}_{2, U_ i}$.

Proof. The implication (1) $\Rightarrow $ (2) is immediate. To see that (2) implies (1) we have to show that every $x$ as in (2) is in the essential image of the functor $F$. To do this choose a covering as in (2), $x_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{1, U_ i})$, and isomorphisms $\varphi _ i : F(x_ i) \to f_ i^*x$. Then we get a descent datum for $\mathcal{S}_1$ relative to $\{ f_ i : U_ i \to U\} $ by taking

\[ \varphi _{ij} : x_ i|_{U_ i \times _ U U_ j} \longrightarrow x_ j|_{U_ i \times _ U U_ j} \]

the arrow such that $F(\varphi _{ij}) = \varphi _ j^{-1} \circ \varphi _ i$. This descent datum is effective by the axioms of a stack, and hence we obtain an object $x_1$ of $\mathcal{S}_1$ over $U$. We omit the verification that $F(x_1)$ is isomorphic to $x$ over $U$. $\square$


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