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 70.24.1. Notation and assumption as in Lemma 70.20.1. The morphism of quotient stacks

\[ [f] : [U/R] \longrightarrow [U'/R'] \]

is fully faithful if and only if $R$ is the restriction of $R'$ via the morphism $f : U \to U'$.

Proof. Let $x, y$ be objects of $[U/R]$ over a scheme $T/S$. Let $x', y'$ be the images of $x, y$ in the category $[U'/R']_ T$. The functor $[f]$ is fully faithful if and only if the map of sheaves

\[ \mathit{Isom}(x, y) \longrightarrow \mathit{Isom}(x', y') \]

is an isomorphism for every $T, x, y$. We may test this locally on $T$ (in the fppf topology). Hence, by Lemma 70.23.1 we may assume that $x, y$ come from $a, b \in U(T)$. In that case we see that $x', y'$ correspond to $f \circ a, f \circ b$. By Lemma 70.21.1 the displayed map of sheaves in this case becomes

\[ T \times _{(a, b), U \times _ B U} R \longrightarrow T \times _{f \circ a, f \circ b, U' \times _ B U'} R'. \]

This is an isomorphism if $R$ is the restriction, because in that case $R = (U \times _ B U) \times _{U' \times _ B U'} R'$, see Lemma 70.16.3 and its proof. Conversely, if the last displayed map is an isomorphism for all $T, a, b$, then it follows that $R = (U \times _ B U) \times _{U' \times _ B U'} R'$, i.e., $R$ is the restriction of $R'$. $\square$


Comments (3)

Comment #1813 by OS on

Formulation of Lemma: is the restriction of (not of )

Comment #3216 by William Chen on

Second line of proof: [U'/'R]_T should be [U'/R']_T


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