The Stacks project

Definition 74.21.1. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. We say $\mathcal{P}$ is étale-smooth local on source-and-target if

1. (stable under precomposing with étale maps) if $f : X \to Y$ is étale and $g : Y \to Z$ has $\mathcal{P}$, then $g \circ f$ has $\mathcal{P}$,

2. (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

3. (locality) given a morphism $f : X \to Y$ the following are equivalent

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

   2. 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 $b$ smooth and $U \to X \times _ Y V$ étale and $u \in |U|$ with $a(u) = x$ such that $h$ has $\mathcal{P}$.