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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

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2 Answers 2

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.

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Equivariant cohomology indeed could be what I want. Leaves open the quest to actually find a space to act on. I have to think about this. –  Tom Bachmann Mar 13 '10 at 9:21
Right now, it appears that your space is G_K itself, so conjugation is a natural choice of action. –  S. Carnahan Mar 13 '10 at 18:43
not sure what you're saying. When I look at the action of G_K on G_K, where are my representations going to come into play? –  Tom Bachmann Mar 13 '10 at 20:36

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).

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Huh. I just saw this question pop up on the front page and thought "This guy asked something quite similar to what I asked a long time ago.". Then I realised it's my own question :D. Thank you for providing the reference, I will look into it. –  Tom Bachmann Apr 11 '12 at 11:57
Seems a funny way to re-discover one's own question :D. Have you made any progress, in the meanwhile? Filippo –  Filippo Alberto Edoardo Apr 11 '12 at 12:12
I didn't really think about it again until today g. –  Tom Bachmann Apr 11 '12 at 17:41

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