Lemma 72.19.3. 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:

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

2. 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 $\mathcal{P}$,

3. 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 $\mathcal{P}$,

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 $U \to X \times _ Y V$ smooth 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 a smooth 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 $i \in I$ 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 72.19.1. The equivalence of (a) and (e) is Lemma 72.19.2 part (2). The equivalence of (a) and (f) is Lemma 72.19.2 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 $\mathcal{P}$. $\square$

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