101.20 Presentations and properties of algebraic stacks
Let (U, R, s, t, c) be a groupoid in algebraic spaces. If s, t : R \to U are flat and locally of finite presentation, then the quotient stack [U/R] is an algebraic stack, see Criteria for Representability, Theorem 97.17.2. In this section we study what properties of (U, R, s, t, c) imply for the algebraic stack [U/R].
Lemma 101.20.1. 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).
If R \to U \times U is separated, then \Delta _\mathcal {X} is separated.
If U, R are separated, then \Delta _\mathcal {X} is separated.
If R \to U \times U is locally quasi-finite, then \mathcal{X} is quasi-DM.
If s, t : R \to U are locally quasi-finite, then \mathcal{X} is quasi-DM.
If R \to U \times U is proper, then \mathcal{X} is separated.
If s, t : R \to U are proper and U is separated, then \mathcal{X} is separated.
Add more here.
Proof.
Observe that the morphism U \to \mathcal{X} is surjective, flat, and locally of finite presentation by Criteria for Representability, Lemma 97.17.1. 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 78.22.2). Thus we see that \Delta _\mathcal {X} has one of the properties listed in Properties of Stacks, Section 100.3 if and only if the morphism R \to U \times U does, see Properties of Stacks, Lemma 100.3.3. 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 67.40.6 hence (6) follows from (5).
\square
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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