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Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:

Points in sites (etale, fppf, ... )

There, Davidac897 gave a nice description of points in more general sites. I would be interested if there is a different description of points in the small fppf site and big etale site over an (as nice as you like) scheme (similar to the very concrete description of points that we have for the small etale site).

Thanks to everyone for all the helpful comments and answers!

The title very much sums it up. The etale site is extremely useful and the basic applications are well known. Milne also devotes time to the Flat site in his Etale Cohomology book. I am hoping that someone can give me example applications.

1.) I'm most interested in the (small) Flat site. What do you typically use this for? Let $X$ a scheme over $\mathbb{F}_P$ and $\alpha\alpha_P$ be the sheaf on the small fppf site over $X$ defined by group scheme $\mathbb{F}_P[t]/(t^p)$. The sequence of sheaves

$$ 0 \rightarrow \alpha\alpha_P \rightarrow \mathbb{G}_a \xrightarrow{F} \mathbb{G}_a \rightarrow 0$$ ($F$ is the map $z \mapsto z^p$) makes sense in the (small) Etale site, but is typically not exact there. However, it becomes exact in the (small) fppf site. This is useful, because a ses of sheaves yields a long exact sequence, and hence relations that one (at least I) cannot so easily express without the Flat site. I don't even know if this example is typical, or if there are many other examples on these lines (or many examples not along these lines).

$\textrm{Principal Homogenous Spaces}$: Cohomology in the flat site calculates the set of principal homogenous spaces over a scheme (wrt a group scheme $G/X$).

2.) What are points in the (small) flat site? I am not able to dream up a good description of these (or where to look).

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    $\begingroup$ A subtle thing that often goes overlooked when dealing with sites finer than the étale is that they are sometimes not functorial. I mean, a map of schemes does not induce a map of the corresponding topos. That's why in SGA4 they switch to big toposes when discussing flat topologies. $\endgroup$
    – Leo Alonso
    Commented Dec 26, 2012 at 15:27
  • $\begingroup$ Leo, You say sometimes - does this difficulty typically manifest in the case $X$ noetherian or should I consider it "non noetherian" phenomena? Thanks for your answer btw. I was wondering why in Milne's book $X_{et}$ denotes the small etale site over $X$ and $X_{fl}$ denotes the big flat site over $X$. $\endgroup$
    – LMN
    Commented Dec 26, 2012 at 17:28
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    $\begingroup$ No, it happens already in very simple cases, e.g. for varieties over a field. The corresponding functor $f^*$ is not exact because the category over which the limit is computed is not filtered. In any case the small flat (fppf) site exists but $f^*$ is only right exact. Perhaps this is what Milne has in mind. $\endgroup$
    – Leo Alonso
    Commented Dec 27, 2012 at 12:25

2 Answers 2

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Briefly, to understand $p$-phenomena in characteristic $p$ you need to replace the etale site by the fppf site. For example, to understand the $p$-torsion in the Brauer group you need the $p$-Kummer sequence and the cohomology of $\mu_{p^n}$, and the study of the $p$-torsion in the Tate-Shafarevich group entails the study of the cohomology of finite group schemes of $p$-power order over curves. There are also many purely geometric applications, e.g., to the Picard functor. You should think of the fppf cohomology as being THE cohomology theory, but the etale topology is fine enough for computing the cohomology of smooth group schemes, and the Zariski topology is fine enough for computing the cohomology of coherent sheaves.

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    $\begingroup$ Thanks for your comments anon! I thought that Crystalline cohomology is "THE" cohomology theory to pick up things in characteristic $p$. Is there an easy comparison you can give between fppf and crystalline cohomology? $\endgroup$
    – LMN
    Commented Dec 28, 2012 at 18:51
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    $\begingroup$ Crystalline cohomology is a Weil cohomology, which explains its importance, but is not very useful for the things I mentioned in my answer. There are relations between flat cohomology and crystalline cohomology, but they are rather complicated to explain. As far as I know, only for the Zariski topology, the etale topology, and topologies in between, do we have a really explicit description of the "points". $\endgroup$
    – anon
    Commented Dec 31, 2012 at 4:41
  • $\begingroup$ anon, Thanks a lot for your comments! $\endgroup$
    – LMN
    Commented Dec 31, 2012 at 4:56
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The cohomology of $\mu_{p^n}$ and $\mathscr{A}[p^n]$ for an Abelian scheme $\mathscr{A}$ in characteristic $p > 0$ can also be studied using syntomic cohomology.

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