The Stacks project

97.4 Pushouts and stacks

In this section we show that algebraic stacks behave well with respect to certain pushouts. The results in this section hold over any base scheme.

The following lemma is also correct when $Y$, $X'$, $X$, $Y'$ are algebraic spaces, see (insert future reference here).


Lemma 97.4.1. 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 37.14.3. 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 93.10.11. 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 80.6.1.

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 $f$. 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 80.6.2 and 80.6.7 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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