Lemma 73.8.2. Any fppf covering is a ph covering, and a fortiori, any syntomic, smooth, étale or Zariski covering is a ph covering.

Proof. We will show that an fppf covering is a ph covering, and then the rest follows from Lemma 73.7.2. Let $\{ X_ i \to X\} _{i \in I}$ be an fppf covering of algebraic spaces over a base scheme $S$. Let $U$ be an affine scheme and let $U \to X$ be a morphism. We can refine the fppf covering $\{ X_ i \times _ U U \to U\} _{i \in I}$ by an fppf covering $\{ T_ i \to U\} _{i \in I}$ where $T_ i$ is a scheme (Lemma 73.7.4). Then we can find a standard ph covering $\{ U_ j \to U\} _{j = 1, \ldots , m}$ refining $\{ T_ i \to U\} _{i \in I}$ by More on Morphisms, Lemma 37.48.7 (and the definition of ph coverings for schemes). Thus $\{ X_ i \to X\} _{i \in I}$ is a ph covering by definition. $\square$

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