The Stacks project

Lemma 101.21.4. 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 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

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

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 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

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

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 ( that $k_0$ is a field.

We claim that
\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 100.11.1 that this morphism is surjective. Thus it suffices to show that ( 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

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

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 100.3.3. $\square$

[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 80.10.1 hence probably should be isolated into its own lemma somewhere.

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