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Suppose that I have a nice variety X over ℚp, with good reduction if you like, and a nice sheaf on X, say coming from a smooth group scheme G. I can cover X by some p-adic open sets Uα, for example the mod-p neighbourhoods coming from some model $\mathcal{X}$ of X. Clearly I can't expect to use Čech cohomology in a naïve way to compute the Hi(X,G) in terms of the cohomology of the Uα, because they don't overlap. But the information about how they fit together to make X is instead contained in the geometry of the special fibre of $\mathcal{X}$.

Is there a spectral sequence which calculates Hi(X,G) in terms of some sort of cohomology of the Uα, and some information about how they fit together?

Motivation

In the situation described above, the Leray spectral sequence gives $$ 0 \to H^1(\mathcal{X},j_*G) \to H^1(X,G) \to H^0(\mathcal{X}, R^1 j_* G)$$ where $j\colon X \to \mathcal{X}$ is the inclusion of the generic fibre.

So in this situation I can compute H1(X,G) in terms of: $R^1 j_* G$, which I want to think of as somehow being the cohomology of the p-adic discs covering X; and the cohomology of $\mathcal{X}$, which I want to think of as saying how those discs fit together.

I would like to see how to generalise this to smaller p-adic neighbourhoods.

Supplementary question:

Should I just go away and read a book on rigid cohomology?

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@Martin: maybe not "just", but rigid cohomology certainly seems relevant to your question. I have a sense that if I went away and read such a book, I would be better equipped to answer it. –  Pete L. Clark Feb 3 '10 at 15:30
    
For what topology on $X$ is your "nice sheaf" a sheaf? Zariski? Etale? What do you mean by $U_{\alpha}$? Are you thinking in terms of the rigid analytification of $X$? –  B. Cais Feb 3 '10 at 18:46
    
I suppose I'm thinking of the étale topology, but I imagine that any answer would apply to other topologies too. I'm thinking of the U_alpha just as p-adic open subsets of X(Q_p), but you're welcome to tell me to put some other structure on them. –  Martin Bright Feb 4 '10 at 11:48

1 Answer 1

up vote 11 down vote accepted

The first comment to make is that Cech theory is really extremely general, and can be set up to compute the cohomology of any complex of abelian sheaves on any site (provided you have coverings that are cohomologically trivial). This is explained at least somewhat in SGA4, Expose 5 and EGA III, Chap 0, section 12.

I think you should be working with the rigid analytic space attached to $X$, and not with the $\mathbf{Q}_p$-points of $X$, and the latter really has no good topology on it besides the totally disconnected one induced from the topology on $\mathbf{Q}_p$.

Let's assume that $X$ has a model $\mathcal{X}$ over $\mathbf{Z}_p$ that is smooth and proper and write $\widehat{\mathcal{X}}$ for the formal completion of $\mathcal{X}$ along its closed fiber. Then the (Berthelot) generic fiber $\widehat{\mathcal{X}}^{rig}$ of $\widehat{\mathcal{X}}$ is a rigid analytic space that is canonically identified with the rigid analytification of $X$ (using properness here). Moreover, one has a "specialization morphism" of ringed sites $$sp:X^{an}\simeq \widehat{\mathcal{X}}^{rig}\rightarrow \widehat{\mathcal{X}}$$ with the property that for any (Zariski) locally closed subset $W$ of the target, the inverse image $sp^{-1}(W)$ is an admissible open of the rigid space $X^{an}$ (called the open tube over W). In this way, coverings of the special fiber by locally closed subsets give coverings of the rigid generic fiber by admissible opens, and you can use Cech theory with these coverings and or your favorite spectral sequence to compute sheaf cohomology in the rigid analytic world. Again using properness, by rigid GAGA this cohomology agrees with usual (Zariski) cohomology on the scheme $X$ (provided your sheaf is a coherent sheaf of $\mathcal{O}_X$-modules, say).

This idea of computing cohomology using admissible coverings of the associated rigid space is a really important one as it allows you to use the geometry of the special fiber. It occurs (allowing $\mathcal{X}$ to have semistable reduction) in the work of Gross on companion forms, of Coleman on $\mathcal{L}$-invariants and most prominently in Iovita-Coleman (see their article on "Frobenius and Monodromy operators"). This latter article might be a good place to start.

I would also highly recommend the articles of Berthelot:

http://perso.univ-rennes1.fr/pierre.berthelot/publis/Cohomologie_Rigide_I.pdf

http://perso.univ-rennes1.fr/pierre.berthelot/publis/Finitude.pdf

I'd also suggest the AWS 2007 notes by Brian Conrad for learning about rigid geometry, which seems generally quite pertinent to your situation. For etale cohomology of rigid spaces, you might want to look at the article of Berkovich, though this would require learning about his analytic spaces.

In any case, I hope this is a good start.

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Does Cech cohomology generalize the notion of a grothendieck topology completely? –  Harry Gindi Feb 4 '10 at 17:34
    
Thank you - this looks very much like what I'm after. Perhaps I should look at those references and then go away and read a book on rigid cohomology! –  Martin Bright Feb 5 '10 at 11:34

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