73.21 Properties of morphisms étale-smooth local on source-and-target

This section is the analogue of Section 73.20 for properties of morphisms which are étale local on the source and smooth local on the target. We give this property a ridiculously long name in order to avoid using it too much.

Definition 73.21.1. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. We say $\mathcal{P}$ is étale-smooth local on source-and-target if

1. (stable under precomposing with étale maps) if $f : X \to Y$ is étale and $g : Y \to Z$ has $\mathcal{P}$, then $g \circ f$ has $\mathcal{P}$,

2. (stable under smooth base change) if $f : X \to Y$ has $\mathcal{P}$ and $Y' \to Y$ is smooth, then the base change $f' : Y' \times _ Y X \to Y'$ has $\mathcal{P}$, and

3. (locality) given a morphism $f : X \to Y$ the following are equivalent

1. $f$ has $\mathcal{P}$,

2. for every $x \in |X|$ there exists a commutative diagram

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

with $b$ smooth and $U \to X \times _ Y V$ étale and $u \in |U|$ with $a(u) = x$ such that $h$ has $\mathcal{P}$.

The above serves as our definition. In the lemmas below we will show that this is equivalent to $\mathcal{P}$ being étale local on the target, smooth local on the source, and stable under post-composing by étale morphisms.

Lemma 73.21.2. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is étale-smooth local on source-and-target. Then

1. $\mathcal{P}$ is étale local on the source,

2. $\mathcal{P}$ is smooth local on the target,

3. $\mathcal{P}$ is stable under postcomposing with étale morphisms: if $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is étale, then $g \circ f$ has $\mathcal{P}$, and

4. $\mathcal{P}$ has a permanence property: given $f : X \to Y$ and $g : Y \to Z$ étale such that $g \circ f$ has $\mathcal{P}$, then $f$ has $\mathcal{P}$.

Proof. We write everything out completely.

Proof of (1). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ X_ i \to X\} _{i \in I}$ be an étale covering of $X$. If each composition $h_ i : X_ i \to Y$ has $\mathcal{P}$, then for each $|x| \in X$ we can find an $i \in I$ and a point $x_ i \in |X_ i|$ mapping to $x$. Then $(X_ i, x_ i) \to (X, x)$ is an étale morphism of pairs, and $\text{id}_ Y : Y \to Y$ is a smooth morphism, and $h_ i$ is as in part (3) of Definition 73.21.1. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$ then each $X_ i \to Y$ has $\mathcal{P}$ by Definition 73.21.1 part (1).

Proof of (2). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ Y_ i \to Y\} _{i \in I}$ be a smooth covering of $Y$. Write $X_ i = Y_ i \times _ Y X$ and $h_ i : X_ i \to Y_ i$ for the base change of $f$. If each $h_ i : X_ i \to Y_ i$ has $\mathcal{P}$, then for each $x \in |X|$ we pick an $i \in I$ and a point $x_ i \in |X_ i|$ mapping to $x$. Then $X_ i \to X \times _ Y Y_ i$ is an étale morphism (because it is an isomorphism), $Y_ i \to Y$ is smooth, and $h_ i$ is as in part (3) of Definition 73.20.1. Thus we see that $f$ has $\mathcal{P}$. Conversely, if $f$ has $\mathcal{P}$, then each $X_ i \to Y_ i$ has $\mathcal{P}$ by Definition 73.20.1 part (2).

Proof of (3). Assume $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is étale. The morphism $X \to Y \times _ Z X$ is étale as a morphism between algebraic spaces étale over $X$ ( Properties of Spaces, Lemma 65.16.6). Also $Y \to Z$ is étale hence a smooth morphism. Thus the diagram

$\xymatrix{ X \ar[d] \ar[r]_ f & Y \ar[d] \\ X \ar[r]^{g \circ f} & Z }$

works for every $x \in |X|$ in part (3) of Definition 73.20.1 and we conclude that $g \circ f$ has $\mathcal{P}$.

Proof of (4). Let $f : X \to Y$ be a morphism and $g : Y \to Z$ étale such that $g \circ f$ has $\mathcal{P}$. Then by Definition 73.21.1 part (2) we see that $\text{pr}_ Y : Y \times _ Z X \to Y$ has $\mathcal{P}$. But the morphism $(f, 1) : X \to Y \times _ Z X$ is étale as a section to the étale projection $\text{pr}_ X : Y \times _ Z X \to X$, see Morphisms of Spaces, Lemma 66.39.11. Hence $f = \text{pr}_ Y \circ (f, 1)$ has $\mathcal{P}$ by Definition 73.21.1 part (1). $\square$

Lemma 73.21.3. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is etale-smooth local on source-and-target. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. $f$ has property $\mathcal{P}$,

2. for every $x \in |X|$ there exists a smooth morphism $b : V \to Y$, an étale morphism $a : U \to V \times _ Y X$, and a point $u \in |U|$ mapping to $x$ such that $U \to V$ has $\mathcal{P}$,

3. for some commutative diagram

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

with $b$ smooth, $U \to V \times _ Y X$ étale, and $a$ surjective the morphism $h$ has $\mathcal{P}$,

4. for any commutative diagram

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

with $b$ smooth and $U \to X \times _ Y V$ étale, the morphism $h$ has $\mathcal{P}$,

5. there exists a smooth covering $\{ Y_ i \to Y\} _{i \in I}$ such that each base change $Y_ i \times _ Y X \to Y_ i$ has $\mathcal{P}$,

6. there exists an étale covering $\{ X_ i \to X\} _{i \in I}$ such that each composition $X_ i \to Y$ has $\mathcal{P}$,

7. there exists a smooth covering $\{ Y_ i \to Y\} _{i \in I}$ and for each $i \in I$ an étale covering $\{ X_{ij} \to Y_ i \times _ Y X\} _{j \in J_ i}$ such that each morphism $X_{ij} \to Y_ i$ has $\mathcal{P}$.

Proof. The equivalence of (a) and (b) is part of Definition 73.21.1. The equivalence of (a) and (e) is Lemma 73.21.2 part (2). The equivalence of (a) and (f) is Lemma 73.21.2 part (1). As (a) is now equivalent to (e) and (f) it follows that (a) equivalent to (g).

It is clear that (c) implies (b). If (b) holds, then for any $x \in |X|$ we can choose a smooth morphism a smooth morphism $b_ x : V_ x \to Y$, an étale morphism $U_ x \to V_ x \times _ Y X$, and $u_ x \in |U_ x|$ mapping to $x$ such that $U_ x \to V_ x$ has $\mathcal{P}$. Then $h = \coprod h_ x : \coprod U_ x \to \coprod V_ x$ with $a = \coprod a_ x$ and $b = \coprod b_ x$ is a diagram as in (c). (Note that $h$ has property $\mathcal{P}$ as $\{ V_ x \to \coprod V_ x\}$ is a smooth covering and $\mathcal{P}$ is smooth local on the target.) Thus (b) is equivalent to (c).

Now we know that (a), (b), (c), (e), (f), and (g) are equivalent. Suppose (a) holds. Let $U, V, a, b, h$ be as in (d). Then $X \times _ Y V \to V$ has $\mathcal{P}$ as $\mathcal{P}$ is stable under smooth base change, whence $U \to V$ has $\mathcal{P}$ as $\mathcal{P}$ is stable under precomposing with étale morphisms. Conversely, if (d) holds, then setting $U = X$ and $V = Y$ we see that $f$ has $\mathcal{P}$. $\square$

Lemma 73.21.4. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume

1. $\mathcal{P}$ is étale local on the source,

2. $\mathcal{P}$ is smooth local on the target, and

3. $\mathcal{P}$ is stable under postcomposing with open immersions: if $f : X \to Y$ has $\mathcal{P}$ and $Y \subset Z$ is an open embedding then $X \to Z$ has $\mathcal{P}$.

Then $\mathcal{P}$ is étale-smooth local on the source-and-target.

Proof. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which satisfies conditions (1), (2) and (3) of the lemma. By Lemma 73.14.2 we see that $\mathcal{P}$ is stable under precomposing with étale morphisms. By Lemma 73.10.2 we see that $\mathcal{P}$ is stable under smooth base change. Hence it suffices to prove part (3) of Definition 73.20.1 holds.

More precisely, suppose that $f : X \to Y$ is a morphism of algebraic spaces over $S$ which satisfies Definition 73.20.1 part (3)(b). In other words, for every $x \in X$ there exists a smooth morphism $b_ x : V_ x \to Y$, an étale morphism $U_ x \to V_ x \times _ Y X$, and a point $u_ x \in |U_ x|$ mapping to $x$ such that $h_ x : U_ x \to V_ x$ has $\mathcal{P}$. The proof of the lemma is complete once we show that $f$ has $\mathcal{P}$.

Let $a_ x : U_ x \to X$ be the composition $U_ x \to V_ x \times _ Y X \to X$. Set $U = \coprod U_ x$, $a = \coprod a_ x$, $V = \coprod V_ x$, $b = \coprod b_ x$, and $h = \coprod h_ x$. We obtain a commutative diagram

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

with $b$ smooth, $U \to V \times _ Y X$ étale, $a$ surjective. Note that $h$ has $\mathcal{P}$ as each $h_ x$ does and $\mathcal{P}$ is smooth local on the target. In the next paragraph we prove that we may assume $U, V, X, Y$ are schemes; we encourage the reader to skip it.

Let $X, Y, U, V, a, b, f, h$ be as in the previous paragraph. We have to show $f$ has $\mathcal{P}$. Let $X' \to X$ be a surjective étale morphism with $X_ i$ a scheme. Set $U' = X' \times _ X U$. Then $U' \to X'$ is surjective and $U' \to X' \times _ Y V$ is étale. Since $\mathcal{P}$ is étale local on the source, we see that $U' \to V$ has $\mathcal{P}$ and that it suffices to show that $X' \to Y$ has $\mathcal{P}$. In other words, we may assume that $X$ is a scheme. Next, choose a surjective étale morphism $Y' \to Y$ with $Y'$ a scheme. Set $V' = V \times _ Y Y'$, $X' = X \times _ Y Y'$, and $U' = U \times _ Y Y'$. Then $U' \to X'$ is surjective and $U' \to X' \times _{Y'} V'$ is étale. Since $\mathcal{P}$ is smooth local on the target, we see that $U' \to V'$ has $\mathcal{P}$ and that it suffices to prove $X' \to Y'$ has $\mathcal{P}$. Thus we may assume both $X$ and $Y$ are schemes. Choose a surjective étale morphism $V' \to V$ with $V'$ a scheme. Set $U' = U \times _ V V'$. Then $U' \to X$ is surjective and $U' \to X \times _ Y V'$ is étale. Since $\mathcal{P}$ is smooth local on the source, we see that $U' \to V'$ has $\mathcal{P}$. Thus we may replace $U, V$ by $U', V'$ and assume $X, Y, V$ are schemes. Finally, we replace $U$ by a scheme surjective étale over $U$ and we see that we may assume $U, V, X, Y$ are all schemes.

If $U, V, X, Y$ are schemes, then $f$ has $\mathcal{P}$ by Descent, Lemma 35.32.11. $\square$

Remark 73.21.5. Using Lemma 73.21.4 and the work done in the earlier sections of this chapter it is easy to make a list of types of morphisms which are smooth local on the source-and-target. In each case we list the lemma which implies the property is etale local on the source and the lemma which implies the property is smooth local on the target. In each case the third assumption of Lemma 73.21.4 is trivial to check, and we omit it. Here is the list:

1. étale, see Lemmas 73.19.1 and 73.11.28,

2. locally quasi-finite, see Lemmas 73.19.2 and 73.11.24,

3. unramified, see Lemmas 73.19.3 and 73.11.27, and

4. add more here as needed.

Of course any property listed in Remark 73.20.5 is a fortiori an example that could be listed here.

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