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Feb 5, 2010 at 11:32 vote accept Martin Bright
Feb 4, 2010 at 13:59 answer added B. Cais timeline score: 14
Feb 4, 2010 at 11:48 comment added Martin Bright 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.
Feb 3, 2010 at 18:46 comment added B. Cais 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 $X$?
Feb 3, 2010 at 15:30 comment added Pete L. Clark @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.
Feb 3, 2010 at 14:52 history asked Martin Bright CC BY-SA 2.5