The Stacks project

Lemma 35.22.4. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{ fpqc, fppf, {\acute{e}tale}, smooth, syntomic\} $. Assume that

  1. the property is preserved under flat, flat and locally of finite presentation, étale, smooth, or syntomic base change depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic (compare with Schemes, Definition 26.18.3),

  2. the property is Zariski local on the base.

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

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

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