Although the definition of etale ($\ell$-adic) cohomology is scary, I have at least some intuition for how it should behave: for instance, when it makes sense, I expect that it should be ``similar'' to the classical (singular) cohomology.

I'm in a situation in which I'm wondering if I can generalize a result in the realm of etale cohomology by using flat cohomology instead. (For instance, I want to be able to consider varieties over characteristic 2 and take cohomology with coefficients in the sheaf $\mu_2$, as an analogue of the case where one of the 2's is replaced by $p \neq 2$.)

Unfortunately, I have no intuition whatsoever for how flat cohomology should behave, so I am looking for mental models, slogans, etc. An answer of "don't use flat cohomology because it is impossible to work with" would also be helpful (although a little disappointing). For the sake of concreteness, here are some specific questions (but my interest is not limited to these)

  1. Can I expect analogues of familiar nice things such as Poincare duality and cycle class maps?
  2. Should this behave like (or even coincide with) etale cohomology in ``nice'' cases?
  3. What are non-trivial examples where flat cohomology can be effectively computed? Especially examples where the computation is not by formally showing that it must be the same as etale cohomology, and then computing the latter.
  4. Should I expect to have a good theory of characteristic classes?
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    $\begingroup$ Some results on flat cohomology can be found in Milne's Arithmetic Duality Theorems, jmilne.org/math/Books/ADTnot.pdf $\endgroup$ – user19475 Mar 2 '16 at 10:37
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    $\begingroup$ For a smooth variety $X$ over a perfect field of characteristic $p$ one can say the following: $H^n_\mathrm{ffp}(X,\mu_p^{\otimes r})\cong H^n_\mathrm{Zar}(X,\mathbb Z/p(r))\cong H^{n-r}_\mathrm{Zar}(X,\nu^r)$, where $\nu^r\subset\Omega^r$ is the etale subsheaf generated by $d\log(f_1)\wedge\dots\wedge d\log(f_r)$. The first isomorphism is more or less the definition of $\mathbb Z/p(r)$ and the second is Geisser-Levine. $\endgroup$ – Marc Hoyois Mar 2 '16 at 16:02
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    $\begingroup$ In particular, you have a cycle class map $CH^r(X)\to H^{2r}_\mathrm{ffp}(X,\mu_p^{\otimes r})$. $\endgroup$ – Marc Hoyois Mar 2 '16 at 16:28
  • $\begingroup$ I don't know much on these things; yet I have heard that syntomic cohomology is much more manageable than the flat one. $\endgroup$ – Mikhail Bondarko Mar 2 '16 at 17:15
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    $\begingroup$ Sorry, I wasn't thinking clearly above. You only have a map $H^n_\mathrm{Zar}(X,\mathbb Z/p(r)) \to H^n_\mathrm{ffp}(X,\mu_p^{\otimes r})$. It's certainly an iso when $n=r=1$. If it's anything like the norm residue isomorphism, you'd only expect it to be an iso for $n\leq r$. Nevertheless, you get cycle classes and characteristic classes from this. $\endgroup$ – Marc Hoyois Mar 2 '16 at 18:01

Here is a good example of some pathological behaviour that will shatter all your hopes and dreams.

Claim. Let $k$ be a perfect field, and let $X$ be a smooth, proper, integral $k$-scheme. Then the cohomology $H^1(X,\alpha_p)$ is a $k^p$-vector space. In particular, it is only finite if it is $0$ or $k$ is finite.

For a proof, see this MO post.

The reason this is disappointing is that $\alpha_p$ should be a constructible sheaf on the flat site, and from étale cohomology we expect the cohomology of a constructible sheaf to be finite. But it isn't!

On the other hand, for $\mu_p$-coefficients, we do have a fairly nice theory, thanks to Milne's arithmetic duality theorems. The first paper in this direction was his Duality in the flat cohomology of a surface, and this has been extended and rewritten many times. Some references were already given in the comments; I am not aware of any others (but they might well exist).

There are many other interesting sheaves to consider other than just sheaves represented by finite group schemes over the ground field. But I don't think there is a nice and unified theory in the same that there is for étale cohomology.

Remark. A nice conceptual remark is the fact that flat cohomology can be computed by the quasi-finite flat site. This is discussed in Milne's étale cohomology book, Example III.3.4. Thus, even though the flat site seems infinitely larger than the étale site (flat morphisms can have arbitrary relative dimension), it somehow isn't as big as you would think.

  • $\begingroup$ Thanks, this is useful. I am actually over a finite field and not working with $\alpha_p$, so this infiniteness doesn't a priori sink my ship, but I guess it is a sign that I shouldn't take any of the usual nice properties for granted. $\endgroup$ – user84144 Mar 4 '16 at 7:56
  • $\begingroup$ Unpromising as the cohomology of $\alpha_p$ is, there is in fact a duality theorem for the cohomology of finite group schemes on curves (due to Artin and Milne). $\endgroup$ – zeno Mar 4 '16 at 23:08

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