The Stacks project

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

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 67.30.8. 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 76.28.1 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

Hence Properties of Stacks, Lemma 100.3.5 implies $\text{pr}_0(U') \cap V(f_1, \ldots , f_ r) \to \mathcal{X}$ is flat and locally of finite presentation as desired. $\square$

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 100.9.20 (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 67.34.6. $\square$

Proof. The implication (2) $\Rightarrow $ (1) is Lemma 101.4.14. 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 101.18.1. Denote $u = \mathop{\mathrm{Spec}}(\kappa (u))$ as usual. Consider the following commutative diagram

with both squares fibre product squares, in particular $R = u \times _\mathcal {X} u$. In the proof of Lemma 101.18.7 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 78.16.2). Note that

As $\mathcal{X}$ is quasi-DM we see that $G$ is locally quasi-finite over $u$. By More on Groupoids in Spaces, Lemma 79.9.11 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

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

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 72.16.1). Hence $\mathcal{O}_{F, \overline{e}}$ is a regular local ring (Properties of Spaces, Lemma 66.25.1). 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 101.21.1 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

By Morphisms of Spaces, Lemma 67.34.1 it follows that $\dim _ e(F_ Z) = 0$ because the transcendence degree of $e$ relative to $u$ is zero. Hence it follows from Lemma 101.21.2 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 100.4.7. By construction the image contains all finite type points of $\mathcal{X}$, hence $f$ is surjective by Lemma 101.18.6 (and Properties of Stacks, Lemma 100.4.4). $\square$

Proof. By Properties of Stacks, Lemma 100.11.3 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 94.9.2. Note that by Algebraic Stacks, Remark 94.16.3 we have an equivalence

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

is unramified over $U$. In particular $\dim (G) = 0$ and by More on Groupoids in Spaces, Lemma 79.9.11 we have $\dim (R) = 0$. This implies that $R$ is a scheme, see Spaces over Fields, Lemma 72.9.1. 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

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

is the desired surjective étale morphism. It follows from Properties of Stacks, Lemma 100.11.1 that this morphism is surjective. Thus it suffices to show that (101.21.4.1) is étale¹. Instead of proving the étaleness directly we first apply Bootstrap, Lemma 80.9.1 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 78.25.2.

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

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 100.3.3. $\square$

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 100.9.20 (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 67.39.12. $\square$

Proof. Recall that (3) is the definition of (2), see Algebraic Stacks, Definition 94.12.2. The implication (3) $\Rightarrow $ (1) is Lemma 101.4.14. 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 101.18.7 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 101.4.16 the algebraic stack $\mathcal{Z}_ x$ is DM. By Lemma 101.21.4 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 67.23.2 and 67.39.9).

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

in other words $F = U \times _{\mathcal{X}, x} \mathop{\mathrm{Spec}}(k)$. By Properties of Stacks, Lemma 100.4.3 we see that $F$ is nonempty. As $\mathcal{Z}_ x \to \mathcal{X}$ is a monomorphism we have

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

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 67.39.11). By Morphisms of Spaces, Lemma 67.51.2 this implies that $\Delta _{F/U}$ is an open immersion, which finally implies by Morphisms of Spaces, Lemma 67.38.9 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

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 67.37.2 and 67.38.3). 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

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 101.21.1. By construction

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

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 101.21.5 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 100.4.7. By construction the image contains all finite type points of $\mathcal{X}$, hence $f$ is surjective by Lemma 101.18.6 (and Properties of Stacks, Lemma 100.4.4). $\square$

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 101.4.4. Since $\mathcal{Z}$ is DM this implies that $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ is DM by Lemma 101.4.11. Hence there exists a scheme $U$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$, see Theorem 101.21.6. Part (2) is a special case of (1) since an algebraic space (when viewed as an algebraic stack) is DM by Lemma 101.4.3. $\square$