The Stacks project

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. $\square$


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