The Stacks project

Proof. Combine Remark 101.34.3 with Morphisms of Spaces, Lemma 67.38.3. $\square$

Proof. Combine Remark 101.34.4 with Morphisms of Spaces, Lemma 67.38.4. $\square$

Proof. Follows from Remark 101.34.5 and Morphisms of Spaces, Lemma 67.39.10. $\square$

Proof. Let $j : \mathcal{Z} \to \mathcal{X}$ be an immersion of algebraic stacks. Since $j$ is representable, it is DM by Lemma 101.4.3. On the other hand, if $X \to \mathcal{X}$ is a smooth and surjective morphism where $X$ is a scheme, then $Z = \mathcal{Z} \times _\mathcal {X} X$ is a locally closed subscheme of $X$. Hence $Z \to X$ is unramified (Morphisms, Lemmas 29.35.7 and 29.35.8) and we conclude that $j$ is unramified from the definition. $\square$

Proof. Assume (1). Then $f$ is DM and since being unramified 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 101.21.7). 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 101.4.5. By Lemma 101.4.12 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 101.21.6.

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 101.21.7). We have to check that $U \to V$ is unramified. It suffices to prove $U \times _\mathcal {Y} W \to V \times _\mathcal {Y} W$ is unramified by Descent on Spaces, Lemma 74.11.27. 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 unramified, and $U$ and $V$ are as before. In this case we know that

by the equivalence of properties (1) and (2) of Lemma 101.34.1 and Definition 101.36.1. This finishes the proof. $\square$

Proof. This follows from Lemma 101.36.6 (characterizing unramified morphisms), Lemma 101.23.7 (characterizing locally quasi-finite morphisms), and Morphisms of Spaces, Lemma 67.38.7 (the corresponding result for algebraic spaces). $\square$

Proof. Assume $\mathcal{X} \to \mathcal{Z}$ is unramified. By Lemma 101.4.12 the morphism $\mathcal{X} \to \mathcal{Y}$ is DM. Choose a commutative diagram

with $U, V, W$ algebraic spaces, with $W \to \mathcal{Z}$ surjective smooth, $V \to \mathcal{Y} \times _\mathcal {Z} W$ surjective étale, and $U \to \mathcal{X} \times _\mathcal {Y} V$ surjective étale (see Lemma 101.21.7). Then also $U \to \mathcal{X} \times _\mathcal {Z} W$ is surjective and étale. Hence we know that $U \to W$ is unramified and we have to show that $U \to V$ is unramified. This follows from Morphisms of Spaces, Lemma 67.38.11. $\square$

Proof. Assume $f$ is unramified. Then $f$ is DM hence we can choose algebraic spaces $U$, $V$, a smooth surjective morphism $V \to \mathcal{Y}$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ (Lemma 101.21.7). Since $f$ is unramified the induced morphism $U \to V$ is unramified. Thus $U \to V$ is locally of finite type (Morphisms of Spaces, Lemma 67.38.6) and we conclude that $f$ is locally of finite type. The diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is a morphism of algebraic stacks over $\mathcal{Y}$. The base change of $\Delta $ by the surjective smooth morphism $V \to \mathcal{Y}$ is the diagonal of the base change of $f$, i.e., of $\mathcal{X}_ V = \mathcal{X} \times _\mathcal {Y} V \to V$. In other words, the diagram

is cartesian. Since the right vertical arrow is surjective and smooth it suffices to show that the top horizontal arrow is étale by Properties of Stacks, Lemma 100.3.4. Consider the commutative diagram

All arrows are representable by algebraic spaces, the vertical arrows are étale, the left one is surjective, and the top horizontal arrow is an open immersion by Morphisms of Spaces, Lemma 67.38.9. This implies what we want: first we see that $U \to \mathcal{X}_ V \times _ V \mathcal{X}_ V$ is étale as a composition of étale morphisms, and then we can use Properties of Stacks, Lemma 100.3.5 to see that $\mathcal{X}_ V \to \mathcal{X}_ V \times _ V \mathcal{X}_ V$ is étale because being étale (for morphisms of algebraic spaces) is local on the source in the étale topology (Descent on Spaces, Lemma 74.19.1).

Assume $f$ is locally of finite type and that its diagonal is étale. Then $f$ is DM by definition (as étale morphisms of algebraic spaces are unramified). As above this means we can choose algebraic spaces $U$, $V$, a smooth surjective morphism $V \to \mathcal{Y}$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ (Lemma 101.21.7). To finish the proof we have to show that $U \to V$ is unramified. We already know that $U \to V$ is locally of finite type. Arguing as above we find a commutative diagram

where all arrows are representable by algebraic spaces, the vertical arrows are étale, and the lower horizontal one is étale as a base change of $\Delta $. It follows that $U \to U \times _ V U$ is étale for example by Lemma 101.35.6¹. Thus $U \to U \times _ V U$ is an étale monomorphism hence an open immersion (Morphisms of Spaces, Lemma 67.51.2). Then $U \to V$ is unramified by Morphisms of Spaces, Lemma 67.38.9. $\square$

Proof. The equivalence of (2) and (3) follows immediately from Lemma 101.36.9. Thus in each case the morphism $f$ is DM. Then we can choose Then we can choose algebraic spaces $U$, $V$, a smooth surjective morphism $V \to \mathcal{Y}$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ (Lemma 101.21.7). To finish the proof we have to show that $U \to V$ is étale if and only if it is locally of finite presentation, flat, and unramified. This follows from Morphisms of Spaces, Lemma 67.39.12 (and the more trivial Morphisms of Spaces, Lemmas 67.39.10, 67.39.8, and 67.39.7). $\square$