I am looking for a general survey on the finite generation properties of
$$H^i(F,\mathbb{Z}_p(j))$$
for fields $F$. Here I refer to Galois cohomology (continuous group cohomology) and the group is defined as the inverse limit over $H^i(F,\mathbb{Z}/p^n(j))$. The only criterion I have seen (and which is already in the work of Tate) is that if $H^i(F,\mathbb{Z}/p^n(j))$ is finite for all $n$, then the above group is a finitely generated $\mathbb{Z}_p$-module.
For example, for finite or local fields this is true, but for number fields it fails (as can, for example, be seen from the Poitou–Tate exact sequence). However, I struggle to find references to discuss other types of fields:
- From the above, I expect that finite generation should be expected to fail for all fields which are finitely generated over the rationals,
- I would expect similarly no finite generation for finitely generated field of transcendence degree at least one over a finite field (using CFT for curves?)
- perhaps finite generation holds for all $d$-local fields, e.g. $\mathbb{Q}_p((t_1))\ldots((t_r))$ for any r?
- and maybe(?) finite generation fails for function fields of varieties of dimension at least one over local fields?
- what about a function field of a curve over an algebraically closed field?
- perhaps something like finite generation could hold over fields like the maximal cyclotomic extension of the rationals (the maximal abelian extension) or maximal extensions ramified only over a single prime?
Most of these are speculations. Does anyone have a good reference to get a good overview? I would similarly like to know about cofiniteness properties of $H^i(F,\mathbb Q_p/\mathbb Z_p(j))$, which I would expect to sometimes just be dual to the above question in the presence of some form of CFT?
