The Stacks project

Lemma 99.21.4. 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 98.11.3 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 92.9.2. Note that by Algebraic Stacks, Remark 92.16.3 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 77.9.11 we have $\dim (R) = 0$. This implies that $R$ is a scheme, see Spaces over Fields, Lemma 70.9.1. 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

99.21.4.1
\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 98.11.1 that this morphism is surjective. Thus it suffices to show that (99.21.4.1) is étale1. Instead of proving the étaleness directly we first apply Bootstrap, Lemma 78.9.1 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 76.24.2.

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 35.20.15, 35.20.11, and 35.20.24. 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 35.20.16.

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 98.3.3. $\square$

[1] 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 78.10.1 hence probably should be isolated into its own lemma somewhere.

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