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)
- Can I expect analogues of familiar nice things such as Poincare duality and cycle class maps?
- Should this behave like (or even coincide with) etale cohomology in ``nice'' cases?
- 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.
- Should I expect to have a good theory of characteristic classes?