The Stacks project

Lemma 92.14.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{Z}$ be a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ whose diagonal is representable by algebraic spaces. Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks over $S$. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of stacks in groupoids. Then the $2$-fibre product $\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack.

Proof. We have to check conditions (1), (2), and (3) of Definition 92.12.1. The first condition follows from Stacks, Lemma 8.5.6.

The second condition we have to check is that the $\mathit{Isom}$-sheaves are representable by algebraic spaces. To do this, suppose that $T$ is a scheme over $S$, and $u, v$ are objects of $(\mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y})_ T$. By our construction of $2$-fibre products (which goes all the way back to Categories, Lemma 4.32.3) we may write $u = (x, y, \alpha )$ and $v = (x', y', \alpha ')$. Here $\alpha : f(x) \to g(y)$ and similarly for $\alpha '$. Then it is clear that

\[ \xymatrix{ \mathit{Isom}(u, v) \ar[d] \ar[rr] & & \mathit{Isom}(y, y') \ar[d]^{\phi \mapsto g(\phi ) \circ \alpha } \\ \mathit{Isom}(x, x') \ar[rr]^-{\psi \mapsto \alpha ' \circ f(\psi )} & & \mathit{Isom}(f(x), g(y')) } \]

is a cartesian diagram of sheaves on $(\mathit{Sch}/T)_{fppf}$. Since by assumption the sheaves $\mathit{Isom}(y, y')$, $\mathit{Isom}(x, x')$, $\mathit{Isom}(f(x), g(y'))$ are algebraic spaces (see Lemma 92.10.11) we see that $\mathit{Isom}(u, v)$ is an algebraic space.

Let $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, and let $x, y$ be surjective smooth morphisms $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$, $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$. Consider the morphism

\[ (\mathit{Sch}/U)_{fppf} \times _{f \circ x, \mathcal{Z}, g \circ y} (\mathit{Sch}/V)_{fppf} \longrightarrow \mathcal{X} \times _{f, \mathcal{Z}, g} \mathcal{Y}. \]

As the diagonal of $\mathcal{Z}$ is representable by algebraic spaces the source of this arrow is representable by an algebraic space $F$, see Lemma 92.10.11. Moreover, the morphism is the composition of base changes of $x$ and $y$, hence surjective and smooth, see Lemmas 92.10.6 and 92.10.5. Choosing a scheme $W$ and a surjective ├ętale morphism $W \to F$ we see that the composition of the displayed $1$-morphism with the corresponding $1$-morphism

\[ (\mathit{Sch}/W)_{fppf} \longrightarrow (\mathit{Sch}/U)_{fppf} \times _{f \circ x, \mathcal{Z}, g \circ y} (\mathit{Sch}/V)_{fppf} \]

is surjective and smooth which proves the last condition. $\square$


Comments (2)

Comment #4854 by SDIGR on

Typo: The morphism in the proof should have source instead of .


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