74.20 Properties of morphisms smooth local on source-and-target
Let $\mathcal{P}$ be a property of morphisms of algebraic spaces. There is an intuitive meaning to the phrase “$\mathcal{P}$ is smooth local on the source and target”. However, it turns out that this notion is not the same as asking $\mathcal{P}$ to be both smooth local on the source and smooth local on the target. We have discussed a similar phenomenon (for the étale topology and the category of schemes) in great detail in Descent, Section 35.32 (for a quick overview take a look at Descent, Remark 35.32.8). However, there is an important difference between the case of the smooth and the étale topology. To see this difference we encourage the reader to ponder the difference between Descent, Lemma 35.32.4 and Lemma 74.20.2 as well as the difference between Descent, Lemma 35.32.5 and Lemma 74.20.3. Namely, in the étale setting the choice of the étale “covering” of the target is immaterial, whereas in the smooth setting it is not.
Definition 74.20.1. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. We say $\mathcal{P}$ is smooth local on source-and-target if
(stable under precomposing with smooth maps) if $f : X \to Y$ is smooth and $g : Y \to Z$ has $\mathcal{P}$, then $g \circ f$ has $\mathcal{P}$,
(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
(locality) given a morphism $f : X \to Y$ the following are equivalent
$f$ has $\mathcal{P}$,
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 smooth vertical arrows 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 smooth local on the target, smooth local on the source, and stable under post-composing by smooth morphisms.
Lemma 74.20.2. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is smooth local on source-and-target. Then
$\mathcal{P}$ is smooth local on the source,
$\mathcal{P}$ is smooth local on the target,
$\mathcal{P}$ is stable under postcomposing with smooth morphisms: if $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is smooth, then $g \circ f$ 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 a smooth 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 a smooth morphism of pairs, and $\text{id}_ Y : Y \to Y$ is a smooth morphism, and $h_ i$ is as in part (3) of Definition 74.20.1. 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 74.20.1 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 $(X_ i, x_ i) \to (X, x)$ is a smooth morphism of pairs, $Y_ i \to Y$ is smooth, and $h_ i$ is as in part (3) of Definition 74.20.1. 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 74.20.1 part (2).
Proof of (3). Assume $f : X \to Y$ has $\mathcal{P}$ and $g : Y \to Z$ is smooth. For every $x \in |X|$ we can think of $(X, x) \to (X, x)$ as a smooth morphism of pairs, $Y \to Z$ is a smooth morphism, and $h = f$ is as in part (3) of Definition 74.20.1. Thus we see that $g \circ f$ has $\mathcal{P}$.
$\square$
The following lemma is the analogue of Morphisms, Lemma 29.14.4.
Lemma 74.20.3. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is smooth local on source-and-target. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:
$f$ has property $\mathcal{P}$,
for every $x \in |X|$ there exists a smooth morphism of pairs $a : (U, u) \to (X, x)$, a smooth morphism $b : V \to Y$, and a morphism $h : U \to V$ such that $f \circ a = b \circ h$ and $h$ has $\mathcal{P}$,
for some commutative diagram
\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]
with $a$, $b$ smooth and $a$ surjective the morphism $h$ has $\mathcal{P}$,
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 $U \to X \times _ Y V$ smooth the morphism $h$ has $\mathcal{P}$,
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 $\mathcal{P}$,
there exists a smooth covering $\{ X_ i \to X\} _{i \in I}$ such that each composition $X_ i \to Y$ has $\mathcal{P}$,
there exists a smooth covering $\{ Y_ i \to Y\} _{i \in I}$ and for each $i \in I$ a smooth 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 74.20.1. The equivalence of (a) and (e) is Lemma 74.20.2 part (2). The equivalence of (a) and (f) is Lemma 74.20.2 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 of pairs $a_ x : (U_ x, u_ x) \to (X, x)$, a smooth morphism $b_ x : V_ x \to Y$, and a morphism $h_ x : U_ x \to V_ x$ such that $f \circ a_ x = b_ x \circ h_ x$ and $h_ x$ 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 smooth morphisms. Conversely, if (d) holds, then setting $U = X$ and $V = Y$ we see that $f$ has $\mathcal{P}$.
$\square$
Lemma 74.20.4. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume
$\mathcal{P}$ is smooth local on the source,
$\mathcal{P}$ is smooth local on the target, and
$\mathcal{P}$ is stable under postcomposing with smooth morphisms: if $f : X \to Y$ has $\mathcal{P}$ and $Y \to Z$ is a smooth morphism then $X \to Z$ has $\mathcal{P}$.
Then $\mathcal{P}$ is 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 74.14.2 we see that $\mathcal{P}$ is stable under precomposing with smooth morphisms. By Lemma 74.10.2 we see that $\mathcal{P}$ is stable under smooth base change. Hence it suffices to prove part (3) of Definition 74.20.1 holds.
More precisely, suppose that $f : X \to Y$ is a morphism of algebraic spaces over $S$ which satisfies Definition 74.20.1 part (3)(b). In other words, for every $x \in X$ there exists a smooth morphism $a_ x : U_ x \to X$, a point $u_ x \in |U_ x|$ mapping to $x$, a smooth morphism $b_ x : V_ x \to Y$, and a morphism $h_ x : U_ x \to V_ x$ such that $f \circ a_ x = b_ x \circ h_ x$ and $h_ x$ has $\mathcal{P}$. The proof of the lemma is complete once we show that $f$ has $\mathcal{P}$. 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 $a$, $b$ smooth, $a$ surjective. Note that $h$ has $\mathcal{P}$ as each $h_ x$ does and $\mathcal{P}$ is smooth local on the target. Because $a$ is surjective and $\mathcal{P}$ is smooth local on the source, it suffices to prove that $b \circ h$ has $\mathcal{P}$. This follows as we assumed that $\mathcal{P}$ is stable under postcomposing with a smooth morphism and as $b$ is smooth.
$\square$
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