Definition 100.35.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is étale if $f$ is DM and the equivalent conditions of Lemma 100.34.1 hold with $\mathcal{P} = {\acute{e}tale}$.
100.35 Étale morphisms
An étale morphism of algebraic stacks should not be defined as a smooth morphism of relative dimension $0$. Namely, the morphism
\[ [\mathbf{A}^1_ k/\mathbf{G}_{m, k}] \longrightarrow \mathop{\mathrm{Spec}}(k) \]
is smooth of relative dimension $0$ for any choice of action of the group scheme $\mathbf{G}_{m, k}$ on $\mathbf{A}^1_ k$. This does not correspond to our usual idea that étale morphisms should identify tangent spaces. The example above isn't quasi-finite, but the morphism
\[ \mathcal{X} = [\mathop{\mathrm{Spec}}(k)/\mu _{p, k}] \longrightarrow \mathop{\mathrm{Spec}}(k) \]
is smooth and quasi-finite (Section 100.23). However, if the characteristic of $k$ is $p > 0$, then we see that the representable morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ isn't étale as the base change $\mu _{p, k} = \mathop{\mathrm{Spec}}(k) \times _\mathcal {X} \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k)$ is a morphism from a nonreduced scheme to the spectrum of a field. Thus if we define an étale morphism as smooth and locally quasi-finite, then the analogue of Morphisms of Spaces, Lemma 66.39.11 would fail.
Instead, our approach will be to start with the requirements that “étaleness” should be a property preserved under base change and that if $\mathcal{X} \to X$ is an étale morphism from an algebraic stack to a scheme, then $\mathcal{X}$ should be Deligne-Mumford. In other words, we will require étale morphisms to be DM and we will use the material in Section 100.34 to define étale morphisms of algebraic stacks.
In Lemma 100.36.10 we will characterize étale morphisms of algebraic stacks as morphisms which are (a) locally of finite presentation, (b) flat, and (c) have étale diagonal.
The property “étale” of morphisms of algebraic spaces is étale-smooth local on the source-and-target, see Descent on Spaces, Remark 73.21.5. It is also stable under base change and fpqc local on the target, see Morphisms of Spaces, Lemma 66.39.4 and Descent on Spaces, Lemma 73.11.28. Hence, by Lemma 100.34.1 above, we may define what it means for a morphism of algebraic spaces to be étale as follows and it agrees with the already existing notion defined in Properties of Stacks, Section 99.3 when the morphism is representable by algebraic spaces because such a morphism is automatically DM by Lemma 100.4.3.
We will use without further mention that this agrees with the already existing notion of étale morphisms in case $f$ is representable by algebraic spaces or if $\mathcal{X}$ and $\mathcal{Y}$ are representable by algebraic spaces.
Lemma 100.35.2. The composition of étale morphisms is étale.
Proof. Combine Remark 100.34.3 with Morphisms of Spaces, Lemma 66.39.3. $\square$
Lemma 100.35.3. A base change of an étale morphism is étale.
Proof. Combine Remark 100.34.4 with Morphisms of Spaces, Lemma 66.39.4. $\square$
Lemma 100.35.4. An open immersion is étale.
Proof. Let $j : \mathcal{U} \to \mathcal{X}$ be an open immersion of algebraic stacks. Since $j$ is representable, it is DM by Lemma 100.4.3. On the other hand, if $X \to \mathcal{X}$ is a smooth and surjective morphism where $X$ is a scheme, then $U = \mathcal{U} \times _\mathcal {X} X$ is an open subscheme of $X$. Hence $U \to X$ is étale (Morphisms, Lemma 29.36.9) and we conclude that $j$ is étale from the definition. $\square$
Lemma 100.35.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent
$f$ is étale,
$f$ is DM and for any morphism $V \to \mathcal{Y}$ where $V$ is an algebraic space and any étale morphism $U \to V \times _\mathcal {Y} \mathcal{X}$ where $U$ is an algebraic space, the morphism $U \to V$ is étale,
there exists some surjective, locally of finite presentation, and flat morphism $W \to \mathcal{Y}$ where $W$ is an algebraic space and some surjective étale morphism $T \to W \times _\mathcal {Y} \mathcal{X}$ where $T$ is an algebraic space such that the morphism $T \to W$ is étale.
Proof. Assume (1). Then $f$ is DM and since being étale is preserved by base change, we see that (2) holds.
Assume (2). Choose a scheme $V$ and a surjective étale morphism $V \to \mathcal{Y}$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _\mathcal {Y} \mathcal{X}$ (Lemma 100.21.7). Thus we see that (3) holds.
Assume $W \to \mathcal{Y}$ and $T \to W \times _\mathcal {Y} \mathcal{X}$ are as in (3). We first check $f$ is DM. Namely, it suffices to check $W \times _\mathcal {Y} \mathcal{X} \to W$ is DM, see Lemma 100.4.5. By Lemma 100.4.12 it suffices to check $W \times _\mathcal {Y} \mathcal{X}$ is DM. This follows from the existence of $T \to W \times _\mathcal {Y} \mathcal{X}$ by (the easy direction of) Theorem 100.21.6.
Assume $f$ is DM and $W \to \mathcal{Y}$ and $T \to W \times _\mathcal {Y} \mathcal{X}$ are as in (3). Let $V$ be an algebraic space, let $V \to \mathcal{Y}$ be surjective smooth, let $U$ be an algebraic space, and let $U \to V \times _\mathcal {Y} \mathcal{X}$ is surjective and étale (Lemma 100.21.7). We have to check that $U \to V$ is étale. It suffices to prove $U \times _\mathcal {Y} W \to V \times _\mathcal {Y} W$ is étale by Descent on Spaces, Lemma 73.11.28. We may replace $\mathcal{X}, \mathcal{Y}, W, T, U, V$ by $\mathcal{X} \times _\mathcal {Y} W, W, W, T, U \times _\mathcal {Y} W, V \times _\mathcal {Y} W$ (small detail omitted). Thus we may assume that $Y = \mathcal{Y}$ is an algebraic space, there exists an algebraic space $T$ and a surjective étale morphism $T \to \mathcal{X}$ such that $T \to Y$ is étale, and $U$ and $V$ are as before. In this case we know that
\[ U \to V\text{ is étale} \Leftrightarrow \mathcal{X} \to Y\text{ is étale} \Leftrightarrow T \to Y\text{ is étale} \]
by the equivalence of properties (1) and (2) of Lemma 100.34.1 and Definition 100.35.1. This finishes the proof. $\square$
Lemma 100.35.6. Let $\mathcal{X}, \mathcal{Y}$ be algebraic stacks étale over an algebraic stack $\mathcal{Z}$. Any morphism $\mathcal{X} \to \mathcal{Y}$ over $\mathcal{Z}$ is étale.
Proof. The morphism $\mathcal{X} \to \mathcal{Y}$ is DM by Lemma 100.4.12. Let $W \to \mathcal{Z}$ be a surjective smooth morphism whose source is an algebraic space. Let $V \to \mathcal{Y} \times _\mathcal {Z} W$ be a surjective étale morphism whose source is an algebraic space (Lemma 100.21.7). Let $U \to \mathcal{X} \times _\mathcal {Y} V$ be a surjective étale morphism whose source is an algebraic space (Lemma 100.21.7). Then
\[ U \longrightarrow \mathcal{X} \times _\mathcal {Z} W \]
is surjective étale as the composition of $U \to \mathcal{X} \times _\mathcal {Y} V$ and the base change of $V \to \mathcal{Y} \times _\mathcal {Z} W$ by $\mathcal{X} \times _\mathcal {Z} W \to \mathcal{Y} \times _\mathcal {Z} W$. Hence it suffices to show that $U \to W$ is étale. Since $U \to W$ and $V \to W$ are étale because $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$ are étale, this follows from the version of the lemma for algebraic spaces, namely Morphisms of Spaces, Lemma 66.39.11. $\square$
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