The classical theory of formal models of rigid analytic spaces due to Raynaud introduces the category of admissible R-formal schemes for $R$ a discretely valued ring, which includes locally topologically of finite type formal R-schemes and so-called special formal R-schemes (covered by formal spectra $\mathrm{Spf} R[[x_1, \ldots, x_n]]/(a_1, \ldots, a_n)$). This category is localised with respect to blowups in the special fibre and a "generic fibre" functor $\eta$ defines an equivalence of categories between the localised category of admissible formal $R$-schemes and the category of rigid analytic spaces over $K$.

It is natural to extend the formal scheme part of the construction to formal schemes over more general base spaces, which was done by Bosch and Luetkebohmert. The resulting generalised category is called by him "relative rigid spaces", and the base space $S$ can be, in particular, an admissible formal $R$-scheme. While the base scheme has a generic fibre in the sense of Raynaud, I couldn't, to my surprise, find a construction of generic fibre functor for relative rigid spaces over $S$. I wonder if I overlooked a well-known fact, or if there is an serious obstacle to such construction. I expect that this fibre functor ought to give rigid analytic $K$-spaces fibred over an analytic space closely related to generic fibre of $S$.

For example, the universal Tate curve is naturally regarded as a relative rigid analytic space over $\mathrm{Spf}R[[q]]$, so the generic fibre of it should give rise to the rigid analytic universal family of totally degenerate elliptic curves over the punctured open disc (punctured is important, as the period $q$ cannot vanish).

My question is, therefore: has the construction of such generic fibre functor of relative rigid analytic spaces over admissible formal $R$-schemes been worked out in the literature?

  • $\begingroup$ Could you give a reference for relative rigid spaces? In your setting, if the base space $S$ is an admissible formal scheme, it seems to me that the existence of the generic fiber functor for spaces over $S$ follows from the funtoriality of the generic fiber functor. $\endgroup$ Jun 16 '17 at 11:42
  • $\begingroup$ By the way, I do not think that the equivalence of categories you wrote works in such a generality. You probably want to consider only formal schemes that are locally topologically of finite type. $\endgroup$ Jun 16 '17 at 11:45
  • $\begingroup$ @JérômePoineau: they are introduced in "Formal and rigid geometry. I. Rigid spaces." Siegfried Bosch; Werner Lütkebohmert Mathematische Annalen (1993) Volume: 295, Issue: 2, page 291-318 I doubt that the answer is immediate from Raynaud's construction, as the example with the universal elliptic curve illustrates. While the relative rigid space is defined over a formal scheme with generic fibre open disc, the rigid analytic space is expected to only be defined over the punctured disc. $\endgroup$ Jun 16 '17 at 11:46
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    $\begingroup$ @DimaSustretov: Generic fibers always exist, you are right (although in this generality, the construction is due to Berthelot and not Raynaud). I was only complaining about the equivalence of categories. Look at the open disk. You get a model of the form Spf $R[[T]]$ but can also get one (much more complicated and certainly not affine) that is locally topologically of finite type. You will not get from one to the other by blowing up and down. $\endgroup$ Jun 16 '17 at 17:23
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    $\begingroup$ @JoeBerner: The generic fiber is defined whatever the setting you want to consider (rigid spaces, Berkovich spaces, Huber spaces). And it is always strictly analytic. $\endgroup$ Jun 16 '17 at 17:25

Yes. You will find what you are looking for in Ahmed Abbes' 'Elements de Geometrie Rigide $I$', where rigid geometry a la Raynaud is developed in a relative setting. It is written in French, but it should cause no trouble to someone who has read the EGA's.

  • $\begingroup$ Dear js21, I have checked the Abbes' book and I found that the treatment there revolves around categories of formal schemes and their localisations. Relative rigid spaces are defined there to be objects of such localised categories, and there doesn't seem to be any functors into the category of rigid analytic space in the classical sense. $\endgroup$ Jun 16 '17 at 13:11
  • $\begingroup$ Is that not how Bosch and Luetkebohmert define relative rigid spaces? I'm looking at page 22 of gdz.sub.uni-goettingen.de/pdfcache/PPN235181684_0295/… $\endgroup$
    – Joe Berner
    Jun 16 '17 at 13:54

If you look at 1.1.12 in Huber's book "Étale cohomology of rigid analytic varieties and adic spaces" he notes that if you embed admissible Noetherian formal schemes into adic spaces, then the "analytic points" functor induces an equivalence of categories after localizing at admissible blowing ups. See the reference for a precise statement.


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