Lemma 78.26.2 (06PC)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.26: Inertia and quotient stacks
Lemma 78.26.2 (cite)

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Lemma 78.26.2. Assume $B \to S$ and $(U, R, s, t, c)$ as in Definition 78.20.1 (1). Let $G/U$ be the stabilizer group algebraic space of the groupoid $(U, R, s, t, c, e, i)$, see Definition 78.16.2. There is a canonical $2$-cartesian diagram

\[ \xymatrix{ \mathcal{S}_ G \ar[r] \ar[d] & \mathcal{S}_ U \ar[d] \\ \mathcal{I}_{[U/R]} \ar[r] & [U/R] } \]

of stacks in groupoids of $(\mathit{Sch}/S)_{fppf}$.

Proof. By Lemma 78.25.3 it suffices to prove that the morphism $s' : R' \to G$ of Lemma 78.26.1 isomorphic to the base change of $s$ by the structure morphism $G \to U$. This base change property is clear from the construction of $s'$. $\square$

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