The Stacks project

101.34 Types of morphisms étale-smooth local on source-and-target

Given a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target, see Descent on Spaces, Definition 74.21.1 we may use it to define a corresponding property of DM morphisms of algebraic stacks, namely by imposing either of the equivalent conditions of the lemma below.

Lemma 101.34.1. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target. Let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks. Consider commutative diagrams

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \]

where $U$ and $V$ are algebraic spaces, $V \to \mathcal{Y}$ is smooth, and $U \to \mathcal{X} \times _\mathcal {Y} V$ is étale. The following are equivalent

  1. for any diagram as above the morphism $h$ has property $\mathcal{P}$, and

  2. for some diagram as above with $a : U \to \mathcal{X}$ surjective the morphism $h$ has property $\mathcal{P}$.

If $\mathcal{X}$ and $\mathcal{Y}$ are representable by algebraic spaces, then this is also equivalent to $f$ (as a morphism of algebraic spaces) having property $\mathcal{P}$. If $\mathcal{P}$ is also preserved under any base change, and fppf local on the base, then for morphisms $f$ which are representable by algebraic spaces this is also equivalent to $f$ having property $\mathcal{P}$ in the sense of Properties of Stacks, Section 100.3.

Proof. Let us prove the implication (1) $\Rightarrow $ (2). Pick an algebraic space $V$ and a surjective and smooth morphism $V \to \mathcal{Y}$. As $f$ is DM there exists a scheme $U$ and a surjective étale morphism $U \to V \times _\mathcal {Y} \mathcal{X}$, see Lemma 101.21.7. Thus we see that (2) holds. Note that $U \to \mathcal{X}$ is surjective and smooth as well, as a composition of the base change $\mathcal{X} \times _\mathcal {Y} V \to \mathcal{X}$ and the chosen map $U \to \mathcal{X} \times _\mathcal {Y} V$. Hence we obtain a diagram as in (1). Thus if (1) holds, then $h : U \to V$ has property $\mathcal{P}$, which means that (2) holds as $U \to \mathcal{X}$ is surjective.

Conversely, assume (2) holds and let $U, V, a, b, h$ be as in (2). Next, let $U', V', a', b', h'$ be any diagram as in (1). Picture

\[ \xymatrix{ U \ar[d] \ar[r]_ h & V \ar[d] \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \quad \quad \xymatrix{ U' \ar[d] \ar[r]_{h'} & V' \ar[d] \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \]

To show that (2) implies (1) we have to prove that $h'$ has $\mathcal{P}$. To do this consider the commutative diagram

\[ \xymatrix{ U \ar[d]^ h & U \times _\mathcal {X} U' \ar[l] \ar[d]^{(h, h')} \ar[r] & U' \ar[d]^{h'} \\ V & V \times _\mathcal {Y} V' \ar[l] \ar[r] & V' } \]

of algebraic spaces. Note that the horizontal arrows are smooth as base changes of the smooth morphisms $V \to \mathcal{Y}$, $V' \to \mathcal{Y}$, $U \to \mathcal{X}$, and $U' \to \mathcal{X}$. Note that the squares

\[ \xymatrix{ U \ar[d] & U \times _\mathcal {X} U' \ar[l] \ar[d] & U \times _\mathcal {X} U' \ar[d] \ar[r] & U' \ar[d] \\ V \times _\mathcal {Y} \mathcal{X} & V \times _\mathcal {Y} U' \ar[l] & U \times _\mathcal {Y} V' \ar[r] & \mathcal{X} \times _\mathcal {Y} V' } \]

are cartesian, hence the vertical arrows are étale by our assumptions on $U', V', a', b', h'$ and $U, V, a, b, h$. Since $\mathcal{P}$ is smooth local on the target by Descent on Spaces, Lemma 74.21.2 part (2) we see that the base change $t : U \times _\mathcal {Y} V' \to V \times _\mathcal {Y} V'$ of $h$ has $\mathcal{P}$. Since $\mathcal{P}$ is étale local on the source by Descent on Spaces, Lemma 74.21.2 part (1) and $s : U \times _\mathcal {X} U' \to U \times _\mathcal {Y} V'$ is étale, we see the morphism $(h, h') = t \circ s$ has $\mathcal{P}$. Consider the diagram

\[ \xymatrix{ U \times _\mathcal {X} U' \ar[r]_{(h, h')} \ar[d] & V \times _\mathcal {Y} V' \ar[d] \\ U' \ar[r]^{h'} & V' } \]

The left vertical arrow is surjective, the right vertical arrow is smooth, and the induced morphism

\[ U \times _\mathcal {X} U' \longrightarrow U' \times _{V'} (V \times _\mathcal {Y} V') = V \times _\mathcal {Y} U' \]

is étale as seen above. Hence by Descent on Spaces, Definition 74.21.1 part (3) we conclude that $h'$ has $\mathcal{P}$. This finishes the proof of the equivalence of (1) and (2).

If $\mathcal{X}$ and $\mathcal{Y}$ are representable, then Descent on Spaces, Lemma 74.21.3 applies which shows that (1) and (2) are equivalent to $f$ having $\mathcal{P}$.

Finally, suppose $f$ is representable, and $U, V, a, b, h$ are as in part (2) of the lemma, and that $\mathcal{P}$ is preserved under arbitrary base change. We have to show that for any scheme $Z$ and morphism $Z \to \mathcal{X}$ the base change $Z \times _\mathcal {Y} \mathcal{X} \to Z$ has property $\mathcal{P}$. Consider the diagram

\[ \xymatrix{ Z \times _\mathcal {Y} U \ar[d] \ar[r] & Z \times _\mathcal {Y} V \ar[d] \\ Z \times _\mathcal {Y} \mathcal{X} \ar[r] & Z } \]

Note that the top horizontal arrow is a base change of $h$ and hence has property $\mathcal{P}$. The left vertical arrow is surjective, the induced morphism

\[ Z \times _\mathcal {Y} U \longrightarrow (Z \times _\mathcal {Y} \mathcal{X}) \times _ Z (Z \times _\mathcal {Y} V) \]

is étale, and the right vertical arrow is smooth. Thus Descent on Spaces, Lemma 74.21.3 kicks in and shows that $Z \times _\mathcal {Y} \mathcal{X} \to Z$ has property $\mathcal{P}$. $\square$

Definition 101.34.2. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target. We say a DM morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks has property $\mathcal{P}$ if the equivalent conditions of Lemma 101.16.1 hold.

Remark 101.34.3. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target and stable under composition. Then the property of DM morphisms of algebraic stacks defined in Definition 101.34.2 is stable under composition. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be DM morphisms of algebraic stacks having property $\mathcal{P}$. By Lemma 101.4.10 the composition $g \circ f$ is DM. Choose an algebraic space $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Choose an algebraic space $V$ and a surjective étale morphism $V \to \mathcal{Y} \times _\mathcal {Z} W$ (Lemma 101.21.7). Choose an algebraic space $U$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$. Then the morphisms $V \to W$ and $U \to V$ have property $\mathcal{P}$ by definition. Whence $U \to W$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under composition. Thus, by definition again, we see that $g \circ f : \mathcal{X} \to \mathcal{Z}$ has property $\mathcal{P}$.

Remark 101.34.4. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is étale-smooth local on the source-and-target and stable under base change. Then the property of DM morphisms of algebraic stacks defined in Definition 101.34.2 is stable under arbitrary base change. Namely, let $f : \mathcal{X} \to \mathcal{Y}$ be a DM morphism of algebraic stacks and $g : \mathcal{Y}' \to \mathcal{Y}$ be a morphism of algebraic stacks and assume $f$ has property $\mathcal{P}$. Then the base change $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ is a DM morphism by Lemma 101.4.4. Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose an algebraic space $U$ and a surjective étale morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ (Lemma 101.21.7). Finally, choose an algebraic space $V'$ and a surjective and smooth morphism $V' \to \mathcal{Y}' \times _\mathcal {Y} V$. Then the morphism $U \to V$ has property $\mathcal{P}$ by definition. Whence $V' \times _ V U \to V'$ has property $\mathcal{P}$ as we assumed that $\mathcal{P}$ is stable under base change. Considering the diagram

\[ \xymatrix{ V' \times _ V U \ar[r] \ar[d] & \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ V' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} } \]

we see that the left top horizontal arrow is surjective and

\[ V' \times _ V U \to V' \times _\mathcal {Y} (\mathcal{Y}' \times _{\mathcal{Y}'} \mathcal{X}) = V' \times _ V (\mathcal{X} \times _\mathcal {Y} V) \]

is étale as a base change of $U \to \mathcal{X} \times _\mathcal {Y} V$, whence by definition we see that the projection $\mathcal{Y}' \times _\mathcal {Y} \mathcal{X} \to \mathcal{Y}'$ has property $\mathcal{P}$.

Remark 101.34.5. Let $\mathcal{P}, \mathcal{P}'$ be properties of morphisms of algebraic spaces which are étale-smooth local on the source-and-target. Suppose that we have $\mathcal{P} \Rightarrow \mathcal{P}'$ for morphisms of algebraic spaces. Then we also have $\mathcal{P} \Rightarrow \mathcal{P}'$ for the properties of morphisms of algebraic stacks defined in Definition 101.34.2 using $\mathcal{P}$ and $\mathcal{P}'$. This is clear from the definition.

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