The Stacks project

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 $\mathcal{P}$. $\square$

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