The Stacks project

Lemma 101.20.2. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces such that $s, t : R \to U$ are flat and locally of finite presentation. Consider the algebraic stack $\mathcal{X} = [U/R]$ (see above). Then the image of $|R| \to |U| \times |U|$ is an equivalence relation and $|\mathcal{X}|$ is the quotient of $|U|$ by this equivalence relation.

Proof. The induced morphism $p : U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation, see Criteria for Representability, Lemma 97.17.1. Hence $|U| \to |\mathcal{X}|$ is surjective by Properties of Stacks, Lemma 100.4.4. Note that $R = U \times _\mathcal {X} U$, see Groupoids in Spaces, Lemma 78.22.2. Hence Properties of Stacks, Lemma 100.4.3 implies the map

\[ |R| \longrightarrow |U| \times _{|\mathcal{X}|} |U| \]

is surjective. Hence the image of $|R| \to |U| \times |U|$ is exactly the set of pairs $(u_1, u_2) \in |U| \times |U|$ such that $u_1$ and $u_2$ have the same image in $|\mathcal{X}|$. Combining these two statements we get the result of the lemma. $\square$

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