Friday, 31 July 2009

ag.algebraic geometry - Cohomology of rigid-analytic spaces

Here's a first pass at your question; hopefully it will suggest something more definitive.



Let's imagine we were in the simplest case, where X is a disk, with its smooth model
being the formal affine line over R, and that Z was the sub-disk of elements of
absolute value less than or equal the absolute value of the uniformizer. Then we can find a semistable model in which Z is one of the covering opens, by blowing up the formal affine line at the origin.



So in this test case, the answer seems to be yes .



Now in general, I think that Raynaud (and/or his collaborators or those who followed in his
tradition) will say that the open immersion ZrightarrowX extends to an open immersion
of formal models. So we can blow up the smooth model of X and the smooth model of Z
so that the latter sits inside the former. What I'm not very certain about is how much
you can control the nature of these blow-ups. (Presumably not at all in general, but you're starting in a fairly nice situation.)



Have you tried asking Brian Conrad yet?

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