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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