ag.algebraic geometry - sheaves of representations on galois groups, can there be interesting cohomology?

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