Comparing topologies

Let S be a scheme. There are many ways to turn the category of schemes over S into a site, but some of the things you can do lead to the same topos, i.e., the category of sheaves are identical. In this case we say these sites define the same topology. Here are some examples of comparisons of topologies:

  1. The smooth topology and the etale topology are the same. See Lemma Tag 055V.
  2. The fppf topology is the same as the one you get by considering fppf coverings {T_i —> T} such that each T_i —> T is locally quasi-finite. See Lemma Tag 0572.
  3. The topology generated by Zariski coverings and {f : T —> S} with f surjective finite locally free is finer than the etale topoloy, see Lemma Tag 02LH and Remark Tag 02LI.
  4. The fppf topology is the same as the one generated by Zariski coverings and finite surjective locally free morphisms. See Lemma Tag 05WN.

Somehow I hadn’t realized 4 earlier. What made me think of it today was this comment by David Rydh.

6 thoughts on “Comparing topologies

  1. This is marvelous (item 3 that is). Is this observation due to Gabber or has it appeared in the literature elsewhere? The universal splitting algebra must be a well-known construction in any case (see the nice section on syntomic morphisms: Algebra 118, tag: 00SR). I was completely unaware of (3), and hence of (4). Has it been used in the literature except in Gabber/Hoobler’s work?

    I also like to complement Johan for the proof of the existence of a standard-├ętale presentation without shrinking the base (if it is affine), Algebra 125.15 (tag: 00UE). This is very nice and I suppose that this adds to the complexity of the proof compared to the usual approach where one can assume that the base is a local ring?

    • Yes, (3) is due to Gabber and is in the paper by Hoobler on Gabber’s proof that Br = Br’ for affine schemes.

      Yes, to your last question (I think). Eventually we should try to improve the presentation. Also, I strongly dislike henselian local rings, and I try to work consistently with étale neighborhoods whenever possible.

      • Well, my question was slightly different. I did look up the lemma in both Gabber’s paper and Hoobler’s but this does not exclude the possibility that it appeared somewhere else before nor that it has appeared (possibly independently) elsewhere afterwards. This was my question.

        Why this aversion towards henselian local rings? Are they worse than local rings? (in another sense than that it could be more rewarding to only work with open neighborhoods/localizations in a single element than with local rings)

        • Yes, sorry, this is possible. I do not know. You might look in the SGA where they introduce all the different topologies on schemes and compare them. (I don’t remember which one this is — I just have a vague memory of what the typography looked like: bad.) Also, you could email Gabber.

          I think it is just that I did a really bad job organizing the section on henselian rings in the stacks project and now it is a bit of a nightmare to use them properly.

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  3. Pingback: Points of the fppf topos | Stacks Project Blog

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