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I have a straightforward question. Let (say) $X/\mathbb{F}_p$ be a smooth proper scheme. On the big crystalline category over $\mathbb{Z}/p^n$ one can take the Zariski or étale topology, and one can define a "small étale-crystalline site" as PD-thickenings $(T,U,\gamma)$ over $(\mathbb{Z}/p^n,\mathbb{F}_p)$ equipped with an étale map $U\to X$. I have the vaguest recollection crystalline theory works fine here and gives basically the same answer, which boils down to the usual comparison theorems between e.g. Zariski and étale quasicoherent sheaves and cohomology. However, basically all references I can find only consider the Zariski topology.

I'm working with stacks for which the fppf topology is the most natural one. There are a few guesses for what the fppf topology should be and it's not obvious which one to take. As well, I worry about whether the formal lifting property of étale morphisms is lurking somewhere in the theory in an essential way.

Does anyone know a reference for a "fppf-crystalline" site or, barring this, an étale-crystalline site?

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    $\begingroup$ I imagine you already checked in Olsson’s Asterisque on crystalline cohomology of stacks? $\endgroup$
    – Piotr Achinger
    Commented Jun 4 at 20:28
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    $\begingroup$ @PiotrAchinger I had not, thank you, though it looks to work with étale-crystalline rather than fppf-crystalline topoi. $\endgroup$
    – Curious
    Commented Jun 4 at 20:35
  • $\begingroup$ Fppf-crystalline could be tricky as typically you use smoothness a lot, and smoothness is not fppf local $\endgroup$
    – Piotr Achinger
    Commented Jun 5 at 6:28
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    $\begingroup$ I do not think that the crystalline cohomology depends on the choice of topology. stacks.math.columbia.edu/tag/07JK basically says that the sheaf cohomology does not depend on the choice of topology, at least for affine schemes, if I understand correctly. They phrase the statement for Zariski, but the proof works for any topology between chaotic and fpqc, by the Serre vanishing of quasicoherent sheaves. $\endgroup$
    – Z. M
    Commented Jun 5 at 11:52
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    $\begingroup$ @PiotrAchinger I do not think that the smoothness is crucial, at least in the approach of Bhatt or maybe de Jong as well, e.g. on the stacks project. $\endgroup$
    – Z. M
    Commented Jun 5 at 11:54

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