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 90.4.1. Let $S$ be a scheme. Let

\[ \xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' } \]

be a pushout in the category of schemes over $S$ where $X \to X'$ is a thickening and $X \to Y$ is affine, see More on Morphisms, Lemma 36.14.3. Let $\mathcal{Z}$ be an algebraic stack over $S$. Then the functor of fibre categories

\[ \mathcal{Z}_{Y'} \longrightarrow \mathcal{Z}_ Y \times _{\mathcal{Z}_ X} \mathcal{Z}_{X'} \]

is an equivalence of categories.

Proof. Let $y'$ be an object of left hand side. The sheaf $\mathit{Isom}(y', y')$ on the category of schemes over $Y'$ is representable by an algebraic space $I$ over $Y'$, see Algebraic Stacks, Lemma 86.10.11. We conclude that the functor of the lemma is fully faithful as $Y'$ is the pushout in the category of algebraic spaces as well as the category of schemes, see Pushouts of Spaces, Lemma 73.2.2.

Let $(y, x', f)$ be an object of the right hand side. Here $f : y|_ X \to x'|_ X$ is an isomorphism. To finish the proof we have to construct an object $y'$ of $\mathcal{Z}_{Y'}$ whose restrictions to $Y$ and $X'$ agree with $y$ and $x'$ in a manner compatible with $\varphi $. In fact, it suffices to construct $y'$ fppf locally on $Y'$, see Stacks, Lemma 8.4.8. Choose a representable algebraic stack $\mathcal{W}$ and a surjective smooth morphism $\mathcal{W} \to \mathcal{Z}$. Then

\[ (\mathit{Sch}/Y)_{fppf} \times _{y, \mathcal{Z}} \mathcal{W} \quad \text{and}\quad (\mathit{Sch}/X')_{fppf} \times _{x', \mathcal{Z}} \mathcal{W} \]

are algebraic stacks representable by algebraic spaces $V$ and $U'$ smooth over $Y$ and $X'$. The isomorphism $f$ induces an isomorphism $\varphi : V \times _ Y X \to U' \times _{X'} X$ over $X$. By Pushouts of Spaces, Lemmas 73.2.4 and 73.2.9 we see that the pushout $V' = V \amalg _{V \times _ Y X} U'$ is an algebraic space smooth over $Y'$ whose base change to $Y$ and $X'$ recovers $V$ and $U'$ in a manner compatible with $\varphi $.

Let $W$ be the algebraic space representing $\mathcal{W}$. The projections $V \to W$ and $U' \to W$ agree as morphisms over $V \times _ Y X \cong U' \times _{X'} X$ hence the universal property of the pushout determines a morphism of algebraic spaces $V' \to W$. Choose a scheme $Y_1'$ and a surjective ├ętale morphism $Y_1' \to V'$. Set $Y_1 = Y \times _{Y'} Y_1'$, $X_1' = X' \times _{Y'} Y_1'$, $X_1 = X \times _{Y'} Y_1'$. The composition

\[ (\mathit{Sch}/Y_1') \to (\mathit{Sch}/V') \to (\mathit{Sch}/W) = \mathcal{W} \to \mathcal{Z} \]

corresponds by the $2$-Yoneda lemma to an object $y_1'$ of $\mathcal{Z}$ over $Y_1'$ whose restriction to $Y_1$ and $X_1'$ agrees with $y|_{Y_1}$ and $x'|_{X_1'}$ in a manner compatible with $f|_{X_1}$. Thus we have constructed our desired object smooth locally over $Y'$ and we win. $\square$


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Suggested slogan: Algebraic stacks satisfy the Rim-Schlessinger condition


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