Lemma 100.20.1. 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).

1. If $R \to U \times U$ is separated, then $\Delta _\mathcal {X}$ is separated.

2. If $U$, $R$ are separated, then $\Delta _\mathcal {X}$ is separated.

3. If $R \to U \times U$ is locally quasi-finite, then $\mathcal{X}$ is quasi-DM.

4. If $s, t : R \to U$ are locally quasi-finite, then $\mathcal{X}$ is quasi-DM.

5. If $R \to U \times U$ is proper, then $\mathcal{X}$ is separated.

6. If $s, t : R \to U$ are proper and $U$ is separated, then $\mathcal{X}$ is separated.

Proof. Observe that the morphism $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation by Criteria for Representability, Lemma 96.17.1. Hence the same is true for $U \times U \to \mathcal{X} \times \mathcal{X}$. We have the cartesian diagram

$\xymatrix{ R = U \times _\mathcal {X} U \ar[r] \ar[d] & U \times U \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X} \times \mathcal{X} }$

(see Groupoids in Spaces, Lemma 77.22.2). Thus we see that $\Delta _\mathcal {X}$ has one of the properties listed in Properties of Stacks, Section 99.3 if and only if the morphism $R \to U \times U$ does, see Properties of Stacks, Lemma 99.3.3. This explains why (1), (3), and (5) are true. The condition in (2) implies $R \to U \times U$ is separated hence (2) follows from (1). The condition in (4) implies the condition in (3) hence (4) follows from (3). The condition in (6) implies the condition in (5) by Morphisms of Spaces, Lemma 66.40.6 hence (6) follows from (5). $\square$

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