Lemma 101.5.7 (06PR)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.5: Inertia stacks
Lemma 101.5.7 (cite)

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Lemma 101.5.7. Let $\mathcal{X}$ be an algebraic stack. Let $[U/R] \to \mathcal{X}$ be a presentation. Let $G/U$ be the stabilizer group algebraic space associated to the groupoid $(U, R, s, t, c)$ . Then

$\xymatrix{ G \ar[d] \ar[r] & U \ar[d] \\ \mathcal{I}_\mathcal {X} \ar[r] & \mathcal{X} }$

is a fibre product diagram.

Proof. Immediate from Groupoids in Spaces, Lemma 78.26.2. $\square$

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