The Stacks project

Lemma in an email of Matthew Emerton dated June 15, 2016

Lemma 92.9.11. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $\mathcal{X} \to \mathcal{Z}$ is representable by algebraic spaces and $\mathcal{Y}$ is a stack in groupoids, then $\mathcal{X} \times _\mathcal {Z} \mathcal{Y}$ is a stack in groupoids.

Proof. The property of a morphism being representable by algebraic spaces is preserved under base-change (Lemma 92.9.8), and so, passing to the base-change $\mathcal{X} \times _\mathcal {Z} \mathcal{Y}$ over $\mathcal{Y}$, we may reduce to the case of a morphism of categories fibred in groupoids $\mathcal{X} \to \mathcal{Y}$ which is representable by algebraic spaces, and whose target is a stack in groupoids; our goal is then to prove that $\mathcal{X}$ is also a stack in groupoids. This follows from Stacks, Lemma 8.6.11 whose assumptions are satisfied as a result of Lemma 92.9.2. $\square$

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