The Stacks project

Lemma 35.26.4. Let $\mathcal{P}$ be a property of morphisms of schemes. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Assume that

1. the property is preserved under precomposing with flat, flat locally of finite presentation, étale, smooth or syntomic morphisms depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic,

2. the property is Zariski local on the source,

3. the property is Zariski local on the target,

4. for any morphism of affine schemes $f : X \to Y$, and any surjective morphism of affine schemes $X' \to X$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic, property $\mathcal{P}$ holds for $f$ if property $\mathcal{P}$ holds for the composition $f' : X' \to Y$.

Then $\mathcal{P}$ is $\tau $ local on the source.