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Although the definition of étale ($\ell$-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 étale 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 Poincaré duality and cycle class maps?
  2. Should this behave like (or even coincide with) étale 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 étale 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
    Commented Mar 2, 2016 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
    Commented Mar 2, 2016 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
    Commented Mar 2, 2016 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
    Commented Mar 2, 2016 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
    Commented Mar 2, 2016 at 18:01

2 Answers 2

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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 way 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.

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  • $\begingroup$ 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
    Commented Mar 4, 2016 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
    Commented Mar 4, 2016 at 23:08
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This is a long remark concerning a new paper by Česnavičius–Scholze on fppf cohomology

Kęstutis Česnavičius. Peter Scholze. Purity for flat cohomology. Ann. of Math. (2) 199 (1) 51 - 180, January 2024. https://doi.org/10.4007/annals.2024.199.1.2

where a purity result of fppf cohomology is established for finite flat commutative group schemes. I am no expert to analyze this result, but let me mention that there is a comparison result between the fppf cohomology and the quasicoherent cohomology:

Theorem. (Thm 4.1.8 in loc. cit.) Let $S$ be a perfect $\mathbb F_p$-scheme and $G$ a commutative, finite, locally free $S$-group that is locally on $S$ of $p$-power order, and let $\epsilon\colon S_{\mathrm{fppf}}\to S_{\mathrm{ét}}$ denote the canonical map of sites. Then we have $$ R\epsilon_*(G)\simeq\operatorname{fib}(\mathbb M(G)\xrightarrow{V-1}\mathbb M(G)) $$ on $S_{\mathrm{ét}}$.

There is also a version for perfectoid $\mathbb Z_p$-schemes as indicated after Thm 1.1.5 in loc. cit., which should follow from applying techniques in §4.2.

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