The Stacks project

Lemma 71.19.4. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Assume

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

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

  3. $\mathcal{P}$ 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 71.13.2 we see that $\mathcal{P}$ is stable under precomposing with smooth morphisms. By Lemma 71.9.2 we see that $\mathcal{P}$ is stable under smooth base change. Hence it suffices to prove part (3) of Definition 71.19.1 holds.

More precisely, suppose that $f : X \to Y$ is a morphism of algebraic spaces over $S$ which satisfies Definition 71.19.1 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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