The Stacks project

Definition 35.32.3. Let $\mathcal{P}$ be a property of morphisms of schemes. We say $\mathcal{P}$ is étale 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 étale base change) if $f : X \to Y$ has $\mathcal{P}$ and $Y' \to Y$ is étale, 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 étale vertical arrows and $u \in U$ with $a(u) = x$ such that $h$ has $\mathcal{P}$.

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