The Stacks project

Lemma 101.3.1. Let $\mathcal{X}$ be an algebraic stack. Let $T$ be a scheme and let $x, y$ be objects of the fibre category of $\mathcal{X}$ over $T$. Then the morphism $\mathit{Isom}_\mathcal {X}(x, y) \to T$ is locally of finite type.

Proof. By Algebraic Stacks, Lemma 94.16.2 we may assume that $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces. By Descent on Spaces, Lemma 74.11.9 it suffices to check the property fppf locally on $T$. Thus we may assume that $x, y$ come from morphisms $x', y' : T \to U$. By Groupoids in Spaces, Lemma 78.22.1 we see that in this case $\mathit{Isom}_\mathcal {X}(x, y) = T \times _{(y', x'), U \times _ S U} R$. Hence it suffices to prove that $R \to U \times _ S U$ is locally of finite type. This follows from the fact that the composition $s : R \to U \times _ S U \to U$ is smooth (hence locally of finite type, see Morphisms of Spaces, Lemmas 67.37.5 and 67.28.5) and Morphisms of Spaces, Lemma 67.23.6. $\square$

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