## 100.21 Special presentations of algebraic stacks

In this section we prove two important theorems. The first is the characterization of quasi-DM stacks $\mathcal{X}$ as the stacks of the form $\mathcal{X} = [U/R]$ with $s, t : R \to U$ locally quasi-finite (as well as flat and locally of finite presentation). The second is the statement that DM algebraic stacks are Deligne-Mumford.

The following lemma gives a criterion for when a “slice” of a presentation is still flat over the algebraic stack.

Lemma 100.21.1. Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram

$\xymatrix{ U \ar[d] & F \ar[l]^ p \ar[d] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l] }$

where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is flat and locally of finite presentation. Let $f_1, \ldots , f_ r \in \Gamma (U, \mathcal{O}_ U)$ and $z \in |F|$ such that $f_1, \ldots , f_ r$ map to a regular sequence in the local ring $\mathcal{O}_{F, \overline{z}}$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism

$V(f_1, \ldots , f_ r) \longrightarrow \mathcal{X}$

is flat and locally of finite presentation.

Proof. Choose a scheme $W$ and a surjective smooth morphism $W \to \mathcal{X}$. Choose an extension of fields $k'/k$ and a morphism $w : \mathop{\mathrm{Spec}}(k') \to W$ such that $\mathop{\mathrm{Spec}}(k') \to W \to \mathcal{X}$ is $2$-isomorphic to $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$. This is possible as $W \to \mathcal{X}$ is surjective. Consider the commutative diagram

$\xymatrix{ U \ar[d] & U \times _\mathcal {X} W \ar[l]^-{\text{pr}_0} \ar[d] & F' \ar[l]^-{p'} \ar[d] \\ \mathcal{X} & W \ar[l] & \mathop{\mathrm{Spec}}(k') \ar[l] }$

both of whose squares are cartesian. By our choice of $w$ we see that $F' = F \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k')$. Thus $F' \to F$ is surjective and we can choose a point $z' \in |F'|$ mapping to $z$. Since $F' \to F$ is flat we see that $\mathcal{O}_{F, \overline{z}} \to \mathcal{O}_{F', \overline{z}'}$ is flat, see Morphisms of Spaces, Lemma 66.30.8. Hence $f_1, \ldots , f_ r$ map to a regular sequence in $\mathcal{O}_{F', \overline{z}'}$, see Algebra, Lemma 10.68.5. Note that $U \times _\mathcal {X} W \to W$ is a morphism of algebraic spaces which is flat and locally of finite presentation. Hence by More on Morphisms of Spaces, Lemma 75.28.1 we see that there exists an open subspace $U'$ of $U \times _\mathcal {X} W$ containing $p(z')$ such that the intersection $U' \cap (V(f_1, \ldots , f_ r) \times _\mathcal {X} W)$ is flat and locally of finite presentation over $W$. Note that $\text{pr}_0(U')$ is an open subspace of $U$ containing $p(z)$ as $\text{pr}_0$ is smooth hence open. Now we see that $U' \cap (V(f_1, \ldots , f_ r) \times _\mathcal {X} W) \to \mathcal{X}$ is flat and locally of finite presentation as the composition

$U' \cap (V(f_1, \ldots , f_ r) \times _\mathcal {X} W) \to W \to \mathcal{X}.$

Hence Properties of Stacks, Lemma 99.3.5 implies $\text{pr}_0(U') \cap V(f_1, \ldots , f_ r) \to \mathcal{X}$ is flat and locally of finite presentation as desired. $\square$

Lemma 100.21.2. Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram

$\xymatrix{ U \ar[d] & F \ar[l]^ p \ar[d] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l] }$

where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is locally of finite type. Let $z \in |F|$ be such that $\dim _ z(F) = 0$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism

$U \longrightarrow \mathcal{X}$

is locally quasi-finite.

Proof. Since $f : U \to \mathcal{X}$ is locally of finite type there exists a maximal open $W(f) \subset U$ such that the restriction $f|_{W(f)} : W(f) \to \mathcal{X}$ is locally quasi-finite, see Properties of Stacks, Remark 99.9.20 (2). Hence all we need to do is prove that $p(z)$ is a point of $W(f)$. Moreover, the remark referenced above also shows the formation of $W(f)$ commutes with arbitrary base change by a morphism which is representable by algebraic spaces. Hence it suffices to show that the morphism $F \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite at $z$. This follows immediately from Morphisms of Spaces, Lemma 66.34.6. $\square$

A quasi-DM stack has a locally quasi-finite “covering” by a scheme.

Theorem 100.21.3. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent

1. $\mathcal{X}$ is quasi-DM, and

2. there exists a scheme $W$ and a surjective, flat, locally finitely presented, locally quasi-finite morphism $W \to \mathcal{X}$.

Proof. The implication (2) $\Rightarrow$ (1) is Lemma 100.4.14. Assume (1). Let $x \in |\mathcal{X}|$ be a finite type point. We will produce a scheme over $\mathcal{X}$ which “works” in a neighbourhood of $x$. At the end of the proof we will take the disjoint union of all of these to conclude.

Let $U$ be an affine scheme, $U \to \mathcal{X}$ a smooth morphism, and $u \in U$ a closed point which maps to $x$, see Lemma 100.18.1. Denote $u = \mathop{\mathrm{Spec}}(\kappa (u))$ as usual. Consider the following commutative diagram

$\xymatrix{ u \ar[d] & R \ar[l] \ar[d] \\ U \ar[d] & F \ar[d] \ar[l]^ p \\ \mathcal{X} & u \ar[l] }$

with both squares fibre product squares, in particular $R = u \times _\mathcal {X} u$. In the proof of Lemma 100.18.7 we have seen that $(u, R, s, t, c)$ is a groupoid in algebraic spaces with $s, t$ locally of finite type. Let $G \to u$ be the stabilizer group algebraic space (see Groupoids in Spaces, Definition 77.16.2). Note that

$G = R \times _{(u \times u)} u = (u \times _\mathcal {X} u) \times _{(u \times u)} u = \mathcal{X} \times _{\mathcal{X} \times \mathcal{X}} u.$

As $\mathcal{X}$ is quasi-DM we see that $G$ is locally quasi-finite over $u$. By More on Groupoids in Spaces, Lemma 78.9.11 we have $\dim (R) = 0$.

Let $e : u \to R$ be the identity of the groupoid. Thus both compositions $u \to R \to u$ are equal to the identity morphism of $u$. Note that $R \subset F$ is a closed subspace as $u \subset U$ is a closed subscheme. Hence we can also think of $e$ as a point of $F$. Consider the maps of étale local rings

$\mathcal{O}_{U, u} \xrightarrow {p^\sharp } \mathcal{O}_{F, \overline{e}} \longrightarrow \mathcal{O}_{R, \overline{e}}$

Note that $\mathcal{O}_{R, \overline{e}}$ has dimension $0$ by the result of the first paragraph. On the other hand, the kernel of the second arrow is $p^\sharp (\mathfrak m_ u)\mathcal{O}_{F, \overline{e}}$ as $R$ is cut out in $F$ by $\mathfrak m_ u$. Thus we see that

$\mathfrak m_{\overline{z}} = \sqrt{p^\sharp (\mathfrak m_ u)\mathcal{O}_{F, \overline{e}}}$

On the other hand, as the morphism $U \to \mathcal{X}$ is smooth we see that $F \to u$ is a smooth morphism of algebraic spaces. This means that $F$ is a regular algebraic space (Spaces over Fields, Lemma 71.16.1). Hence $\mathcal{O}_{F, \overline{e}}$ is a regular local ring (Properties of Spaces, Lemma 65.25.1). Note that a regular local ring is Cohen-Macaulay (Algebra, Lemma 10.106.3). Let $d = \dim (\mathcal{O}_{F, \overline{e}})$. By Algebra, Lemma 10.104.10 we can find $f_1, \ldots , f_ d \in \mathcal{O}_{U, u}$ whose images $\varphi (f_1), \ldots , \varphi (f_ d)$ form a regular sequence in $\mathcal{O}_{F, \overline{z}}$. By Lemma 100.21.1 after shrinking $U$ we may assume that $Z = V(f_1, \ldots , f_ d) \to \mathcal{X}$ is flat and locally of finite presentation. Note that by construction $F_ Z = Z \times _\mathcal {X} u$ is a closed subspace of $F = U \times _\mathcal {X} u$, that $e$ is a point of this closed subspace, and that

$\dim (\mathcal{O}_{F_ Z, \overline{e}}) = 0.$

By Morphisms of Spaces, Lemma 66.34.1 it follows that $\dim _ e(F_ Z) = 0$ because the transcendence degree of $e$ relative to $u$ is zero. Hence it follows from Lemma 100.21.2 that after possibly shrinking $U$ the morphism $Z \to \mathcal{X}$ is locally quasi-finite.

We conclude that for every finite type point $x$ of $\mathcal{X}$ there exists a locally quasi-finite, flat, locally finitely presented morphism $f_ x : Z_ x \to \mathcal{X}$ with $x$ in the image of $|f_ x|$. Set $W = \coprod _ x Z_ x$ and $f = \coprod f_ x$. Then $f$ is flat, locally of finite presentation, and locally quasi-finite. In particular the image of $|f|$ is open, see Properties of Stacks, Lemma 99.4.7. By construction the image contains all finite type points of $\mathcal{X}$, hence $f$ is surjective by Lemma 100.18.6 (and Properties of Stacks, Lemma 99.4.4). $\square$

Lemma 100.21.4. Let $\mathcal{Z}$ be a DM, locally Noetherian, reduced algebraic stack with $|\mathcal{Z}|$ a singleton. Then there exists a field $k$ and a surjective étale morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$.

Proof. By Properties of Stacks, Lemma 99.11.3 there exists a field $k$ and a surjective, flat, locally finitely presented morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{Z}$. Set $U = \mathop{\mathrm{Spec}}(k)$ and $R = U \times _\mathcal {Z} U$ so we obtain a groupoid in algebraic spaces $(U, R, s, t, c)$, see Algebraic Stacks, Lemma 93.9.2. Note that by Algebraic Stacks, Remark 93.16.3 we have an equivalence

$f_{can} : [U/R] \longrightarrow \mathcal{Z}$

The projections $s, t : R \to U$ are locally of finite presentation. As $\mathcal{Z}$ is DM we see that the stabilizer group algebraic space

$G = U \times _{U \times U} R = U \times _{U \times U} (U \times _\mathcal {Z} U) = U \times _{\mathcal{Z} \times \mathcal{Z}, \Delta _\mathcal {Z}} \mathcal{Z}$

is unramified over $U$. In particular $\dim (G) = 0$ and by More on Groupoids in Spaces, Lemma 78.9.11 we have $\dim (R) = 0$. This implies that $R$ is a scheme, see Spaces over Fields, Lemma 71.9.1. By Varieties, Lemma 33.20.2 we see that $R$ (and also $G$) is the disjoint union of spectra of Artinian local rings finite over $k$ via either $s$ or $t$. Let $P = \mathop{\mathrm{Spec}}(A) \subset R$ be the open and closed subscheme whose underlying point is the identity $e$ of the groupoid scheme $(U, R, s, t, c)$. As $s \circ e = t \circ e = \text{id}_{\mathop{\mathrm{Spec}}(k)}$ we see that $A$ is an Artinian local ring whose residue field is identified with $k$ via either $s^\sharp : k \to A$ or $t^\sharp : k \to A$. Note that $s, t : \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(k)$ are finite (by the lemma referenced above). Since $G \to \mathop{\mathrm{Spec}}(k)$ is unramified we see that

$G \cap P = P \times _{U \times U} U = \mathop{\mathrm{Spec}}(A \otimes _{k \otimes k} k)$

is unramified over $k$. On the other hand $A \otimes _{k \otimes k} k$ is local as a quotient of $A$ and surjects onto $k$. We conclude that $A \otimes _{k \otimes k} k = k$. It follows that $P \to U \times U$ is universally injective (as $P$ has only one point with residue field $k$), unramified (by the computation of the fibre over the unique image point above), and of finite type (because $s, t$ are) hence a monomorphism (see Étale Morphisms, Lemma 41.7.1). Thus $s|_ P, t|_ P : P \to U$ define a finite flat equivalence relation. Thus we may apply Groupoids, Proposition 39.23.9 to conclude that $U/P$ exists and is a scheme $\overline{U}$. Moreover, $U \to \overline{U}$ is finite locally free and $P = U \times _{\overline{U}} U$. In fact $\overline{U} = \mathop{\mathrm{Spec}}(k_0)$ where $k_0 \subset k$ is the ring of $R$-invariant functions. As $k$ is a field it follows from the definition Groupoids, Equation (39.23.0.1) that $k_0$ is a field.

We claim that

100.21.4.1
\begin{equation} \label{stacks-morphisms-equation-etale-covering} \mathop{\mathrm{Spec}}(k_0) = \overline{U} = U/P \to [U/R] = \mathcal{Z} \end{equation}

is the desired surjective étale morphism. It follows from Properties of Stacks, Lemma 99.11.1 that this morphism is surjective. Thus it suffices to show that (100.21.4.1) is étale1. Instead of proving the étaleness directly we first apply Bootstrap, Lemma 79.9.1 to see that there exists a groupoid scheme $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ such that $(U, R, s, t, c)$ is the restriction of $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ via the quotient morphism $U \to \overline{U}$. (We verified all the hypothesis of the lemma above except for the assertion that $j : R \to U \times U$ is separated and locally quasi-finite which follows from the fact that $R$ is a separated scheme locally quasi-finite over $k$.) Since $U \to \overline{U}$ is finite locally free we see that $[U/R] \to [\overline{U}/\overline{R}]$ is an equivalence, see Groupoids in Spaces, Lemma 77.25.2.

Note that $s, t$ are the base changes of the morphisms $\overline{s}, \overline{t}$ by $U \to \overline{U}$. As $\{ U \to \overline{U}\}$ is an fppf covering we conclude $\overline{s}, \overline{t}$ are flat, locally of finite presentation, and locally quasi-finite, see Descent, Lemmas 35.23.15, 35.23.11, and 35.23.24. Consider the commutative diagram

$\xymatrix{ U \times _{\overline{U}} U \ar@{=}[r] \ar[rd] & P \ar[r] \ar[d] & R \ar[d] \\ & \overline{U} \ar[r]^{\overline{e}} & \overline{R} }$

It is a general fact about restrictions that the outer four corners form a cartesian diagram. By the equality we see the inner square is cartesian. Since $P$ is open in $R$ we conclude that $\overline{e}$ is an open immersion by Descent, Lemma 35.23.16.

But of course, if $\overline{e}$ is an open immersion and $\overline{s}, \overline{t}$ are flat and locally of finite presentation then the morphisms $\overline{t}, \overline{s}$ are étale. For example you can see this by applying More on Groupoids, Lemma 40.4.1 which shows that $\Omega _{\overline{R}/\overline{U}} = 0$ implies that $\overline{s}, \overline{t} : \overline{R} \to \overline{U}$ is unramified (see Morphisms, Lemma 29.35.2), which in turn implies that $\overline{s}, \overline{t}$ are étale (see Morphisms, Lemma 29.36.16). Hence $\mathcal{Z} = [\overline{U}/\overline{R}]$ is an étale presentation of the algebraic stack $\mathcal{Z}$ and we conclude that $\overline{U} \to \mathcal{Z}$ is étale by Properties of Stacks, Lemma 99.3.3. $\square$

Lemma 100.21.5. Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram

$\xymatrix{ U \ar[d] & F \ar[l]^ p \ar[d] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l] }$

where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is flat and locally of finite presentation. Let $z \in |F|$ be such that $F \to \mathop{\mathrm{Spec}}(k)$ is unramified at $z$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism

$U \longrightarrow \mathcal{X}$

is étale.

Proof. Since $f : U \to \mathcal{X}$ is flat and locally of finite presentation there exists a maximal open $W(f) \subset U$ such that the restriction $f|_{W(f)} : W(f) \to \mathcal{X}$ is étale, see Properties of Stacks, Remark 99.9.20 (5). Hence all we need to do is prove that $p(z)$ is a point of $W(f)$. Moreover, the remark referenced above also shows the formation of $W(f)$ commutes with arbitrary base change by a morphism which is representable by algebraic spaces. Hence it suffices to show that the morphism $F \to \mathop{\mathrm{Spec}}(k)$ is étale at $z$. Since it is flat and locally of finite presentation as a base change of $U \to \mathcal{X}$ and since $F \to \mathop{\mathrm{Spec}}(k)$ is unramified at $z$ by assumption, this follows from Morphisms of Spaces, Lemma 66.39.12. $\square$

A DM stack is a Deligne-Mumford stack.

Theorem 100.21.6. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent

1. $\mathcal{X}$ is DM,

2. $\mathcal{X}$ is Deligne-Mumford, and

3. there exists a scheme $W$ and a surjective étale morphism $W \to \mathcal{X}$.

Proof. Recall that (3) is the definition of (2), see Algebraic Stacks, Definition 93.12.2. The implication (3) $\Rightarrow$ (1) is Lemma 100.4.14. Assume (1). Let $x \in |\mathcal{X}|$ be a finite type point. We will produce a scheme over $\mathcal{X}$ which “works” in a neighbourhood of $x$. At the end of the proof we will take the disjoint union of all of these to conclude.

By Lemma 100.18.7 the residual gerbe $\mathcal{Z}_ x$ of $\mathcal{X}$ at $x$ exists and $\mathcal{Z}_ x \to \mathcal{X}$ is locally of finite type. By Lemma 100.4.16 the algebraic stack $\mathcal{Z}_ x$ is DM. By Lemma 100.21.4 there exists a field $k$ and a surjective étale morphism $z : \mathop{\mathrm{Spec}}(k) \to \mathcal{Z}_ x$. In particular the composition $x : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ is locally of finite type (by Morphisms of Spaces, Lemmas 66.23.2 and 66.39.9).

Pick a scheme $U$ and a smooth morphism $U \to \mathcal{X}$ such that $x$ is in the image of $|U| \to |\mathcal{X}|$. Consider the following fibre square

$\xymatrix{ U \ar[d] & F \ar[l] \ar[d] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l]_-x }$

in other words $F = U \times _{\mathcal{X}, x} \mathop{\mathrm{Spec}}(k)$. By Properties of Stacks, Lemma 99.4.3 we see that $F$ is nonempty. As $\mathcal{Z}_ x \to \mathcal{X}$ is a monomorphism we have

$\mathop{\mathrm{Spec}}(k) \times _{z, \mathcal{Z}_ x, z} \mathop{\mathrm{Spec}}(k) = \mathop{\mathrm{Spec}}(k) \times _{x, \mathcal{X}, x} \mathop{\mathrm{Spec}}(k)$

with étale projection maps to $\mathop{\mathrm{Spec}}(k)$ by construction of $z$. Since

$F \times _ U F = (\mathop{\mathrm{Spec}}(k) \times _\mathcal {X} \mathop{\mathrm{Spec}}(k)) \times _{\mathop{\mathrm{Spec}}(k)} F$

we see that the projections maps $F \times _ U F \to F$ are étale as well. It follows that $\Delta _{F/U} : F \to F \times _ U F$ is étale (see Morphisms of Spaces, Lemma 66.39.11). By Morphisms of Spaces, Lemma 66.51.2 this implies that $\Delta _{F/U}$ is an open immersion, which finally implies by Morphisms of Spaces, Lemma 66.38.9 that $F \to U$ is unramified.

Pick a nonempty affine scheme $V$ and an étale morphism $V \to F$. (This could be avoided by working directly with $F$, but it seems easier to explain what's going on by doing so.) Picture

$\xymatrix{ U \ar[d] & F \ar[l] \ar[d] & V \ar[l] \ar[ld] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l]_-x }$

Then $V \to \mathop{\mathrm{Spec}}(k)$ is a smooth morphism of schemes and $V \to U$ is an unramified morphism of schemes (see Morphisms of Spaces, Lemmas 66.37.2 and 66.38.3). Pick a closed point $v \in V$ with $k \subset \kappa (v)$ finite separable, see Varieties, Lemma 33.25.6. Let $u \in U$ be the image point. The local ring $\mathcal{O}_{V, v}$ is regular (see Varieties, Lemma 33.25.3) and the local ring homomorphism

$\varphi : \mathcal{O}_{U, u} \longrightarrow \mathcal{O}_{V, v}$

coming from the morphism $V \to U$ is such that $\varphi (\mathfrak m_ u)\mathcal{O}_{V, v} = \mathfrak m_ v$, see Morphisms, Lemma 29.35.14. Hence we can find $f_1, \ldots , f_ d \in \mathcal{O}_{U, u}$ such that the images $\varphi (f_1), \ldots , \varphi (f_ d)$ form a basis for $\mathfrak m_ v/\mathfrak m_ v^2$ over $\kappa (v)$. Since $\mathcal{O}_{V, v}$ is a regular local ring this implies that $\varphi (f_1), \ldots , \varphi (f_ d)$ form a regular sequence in $\mathcal{O}_{V, v}$ (see Algebra, Lemma 10.106.3). After replacing $U$ by an open neighbourhood of $u$ we may assume $f_1, \ldots , f_ d \in \Gamma (U, \mathcal{O}_ U)$. After replacing $U$ by a possibly even smaller open neighbourhood of $u$ we may assume that $V(f_1, \ldots , f_ d) \to \mathcal{X}$ is flat and locally of finite presentation, see Lemma 100.21.1. By construction

$V(f_1, \ldots , f_ d) \times _\mathcal {X} \mathop{\mathrm{Spec}}(k) \longleftarrow V(f_1, \ldots , f_ d) \times _ U V$

is étale and $V(f_1, \ldots , f_ d) \times _ U V$ is the closed subscheme $T \subset V$ cut out by $f_1|_ V, \ldots , f_ d|_ V$. Hence by construction $v \in T$ and

$\mathcal{O}_{T, v} = \mathcal{O}_{V, v}/(\varphi (f_1), \ldots , \varphi (f_ d)) = \kappa (v)$

a finite separable extension of $k$. It follows that $T \to \mathop{\mathrm{Spec}}(k)$ is unramified at $v$, see Morphisms, Lemma 29.35.14. By definition of an unramified morphism of algebraic spaces this means that $V(f_1, \ldots , f_ d) \times _\mathcal {X} \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k)$ is unramified at the image of $v$ in $V(f_1, \ldots , f_ d) \times _\mathcal {X} \mathop{\mathrm{Spec}}(k)$. Applying Lemma 100.21.5 we see that on shrinking $U$ to yet another open neighbourhood of $u$ the morphism $V(f_1, \ldots , f_ d) \to \mathcal{X}$ is étale.

We conclude that for every finite type point $x$ of $\mathcal{X}$ there exists an étale morphism $f_ x : W_ x \to \mathcal{X}$ with $x$ in the image of $|f_ x|$. Set $W = \coprod _ x W_ x$ and $f = \coprod f_ x$. Then $f$ is étale. In particular the image of $|f|$ is open, see Properties of Stacks, Lemma 99.4.7. By construction the image contains all finite type points of $\mathcal{X}$, hence $f$ is surjective by Lemma 100.18.6 (and Properties of Stacks, Lemma 99.4.4). $\square$

Here is a useful corollary which tells us that the “fibres” of a DM morphism of algebraic stacks are Deligne-Mumford.

Lemma 100.21.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks. Then

1. For every DM algebraic stack $\mathcal{Z}$ and morphism $\mathcal{Z} \to \mathcal{Y}$ there exists a scheme and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$.

2. For every algebraic space $Z$ and morphism $Z \to \mathcal{Y}$ there exists a scheme and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} Z$.

Proof. Proof of (1). As $f$ is DM we see that the base change $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{Z}$ is DM by Lemma 100.4.4. Since $\mathcal{Z}$ is DM this implies that $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ is DM by Lemma 100.4.11. Hence there exists a scheme $U$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$, see Theorem 100.21.6. Part (2) is a special case of (1) since an algebraic space (when viewed as an algebraic stack) is DM by Lemma 100.4.3. $\square$

 We urge the reader to find his/her own proof of this fact. In fact the argument has a lot in common with the final argument of the proof of Bootstrap, Theorem 79.10.1 hence probably should be isolated into its own lemma somewhere.

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