Lemma 97.12.6 (05YJ)—The Stacks project

Lemma 97.12.6. Let $F : \mathcal{X} \to \mathcal{Y}$ and $G : \mathcal{X}' \to \mathcal{X}$ be $1$ -morphisms of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$ . If $G$ is representable by algebraic spaces, then the $1$ -morphism

$\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}) \longrightarrow \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$

is representable by algebraic spaces.

Proof. Let $U$ be a scheme and let $\xi = (U, Z, y, x, \alpha )$ be an object of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ over $U$ . We have to prove that the $2$ -fibre product

97.12.6.1

$\begin{equation} \label{criteria-equation-to-show-again} (\mathit{Sch}/U)_{fppf} \times _{\xi , \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})} \mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}) \end{equation}$

is representable by an algebraic space étale over $U$ . An object of this over $a : U' \to U$ corresponds to an object $x'$ of $\mathcal{X}'$ over $U' \times _{a, U} Z$ such that $G(x') \cong x|_{U' \times _{a, U} Z}$ . By assumption the $2$ -fibre product

$(\mathit{Sch}/Z)_{fppf} \times _{x, \mathcal{X}} \mathcal{X}'$

is representable by an algebraic space $X$ over $Z$ . It follows that (97.12.6.1) is representable by $\text{Res}_{Z/U}(X)$ , which is an algebraic space by Proposition 97.11.5. $\square$

The Stacks project

Comments (0)

Post a comment