Lemma 34.10.10 (0ETK)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 34: Topologies on Schemes
Section 34.10: The V topology
Lemma 34.10.10 (cite)

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Lemma 34.10.10. Any fpqc covering is a V covering. A fortiori, any fppf, syntomic, smooth, étale or Zariski covering is a V covering. Also, a ph covering is a V covering.

Proof. An fpqc covering can affine locally be refined by a standard fpqc covering, see Lemmas 34.9.9. A standard fpqc covering is a standard V covering, see Lemma 34.10.2. Hence the first statement follows from our definition of V covers in terms of standard V coverings. The conclusion for fppf, syntomic, smooth, étale or Zariski coverings follows as these are fpqc coverings, see Lemma 34.9.7.

The statement on ph coverings follows from Lemma 34.10.3 in the same manner. $\square$

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