sheaves of representations on galois groups, can there be interesting cohomology?

Question

Consider a field $K$ (of characteristic 0, say) and its absolute galois group $G_K^{ab} = Gal(\overline{K}/K)$, given the Krull topology: $U_E(\sigma) = \sigma Gal(\overline{K}/E)$ form a basis of the topology, ranging over $\sigma \in G_K^{ab}$ and $E/K$ finite galois.

Fix a group $G$ and denote by $R_E$ its representation ring over $E$, and by $R_E^\sigma \subset R_E$ the elements of $R_E$ fixed by $\sigma$.

We can construct a sheaf $\mathcal{F}$ on $G_K^{ab}$ by setting $\mathcal{F}(U_E(\sigma)) = R_E^\sigma$. It is a simple exercise to verify the axioms.

One might hope that the sheaf cohomology of $\mathcal{F}$ encodes information about the splitting behaviour of representations of G over various ground fields, but this is not the case: $G_K^{ab}$ is known to be totally disconnected, hausdorff and compact. It is a theorem [1, 5.1] that $H^r(G_K^{ab}, \mathcal{F}) = 0$ for $r > 0$. Furthermore the $U_E(\sigma)$ are actually clopen, so most useful subsets I can think of are also compact, hence their cohomology is equally uninteresting.

Is there a way to produce a useful cohomology along these lines?

Here "useful" essentially means "non-trivial", and "along these lines" basically "involving the galois action on $R_E$ for various $E$".

[1] http://www.jstor.org/stable/2035693

Answer 1

This is just a shot in the dark, but you might want to consider $G_K$-equivariant cohomology, with conjugation action on the space. At the very least, the equivariance will add some contribution from the group cohomology.

Answer 2

Have you tried looking at Iwasawa's paper "Sheaves over Number Fields", http://www.jstor.org/stable/1970190?

He does not really do what you would like to, because his base space are the divisors of the number field, with some funny topology. But by class field theory it could give examples on the push-forward of your sheaf on the maximal abelian quotient of your Galois group (I am assuming your field is a global one, which might be out of your interest - sorry).