The Stacks project

Lemma 35.23.4. 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 $X \to Y$, 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.

Proof. This follows almost immediately from the definition of a $\tau $-covering, see Topologies, Definition 34.9.1 34.7.1 34.4.1 34.5.1, or 34.6.1 and Topologies, Lemma 34.9.8, 34.7.4, 34.4.4, 34.5.4, or 34.6.4. Details omitted. (Hint: Use locality on the source and target to reduce the verification of property $\mathcal{P}$ to the case of a morphism between affines. Then apply (1) and (4).) $\square$


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