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
Comments (0)