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Frédéric, this sounds like a very good idea – I hope you continue adding to it.
Frédéric, there is an old stub rigid analytic geometry, which seem to be essentially about the same subject. I am familiar with rigid terminology, but not with “global” terminology. is this restriction to the case of global fields or what ? It is unfortunate terminology as in differential geometry global differential geometry is meants as a specific area of differential geometry.
Well, I do not know what do you mean by “usual rigid analytic geometry”. People whom I know classify under this name plethora of modern variants including e.g. by Huber and I think Berkovich spectra are also included.
Come on, rarely anybody works with the first version due Tate. If one restricts the name “rigid geometry” to his version then one may come in conflict with the practice of most active practioners. OK, I am happy with saying classical analytic spaces for the Tate version, but I continue to call the whole wide subject rigid analytic geometry. The name of Poineau is completely new to me. Thanks for pointing out. Somebody participating in a cycle of seminars at IHES explained to me 7 years ago about rigid geometry and they were working Huber version.
When mentioning places at global analytic geometry, would you be so kind to write an entry for place ? I wanted to write one but felt incompetent.
I agree that the subject is unique; for example it draws much to the classical material in
The classical results like Weierstrass division, Weierstrass preparation and so on, hold over many fields; and one sees already there the role of Banach algebras etc. Now the Tate’s affinoid picture and usage of Grothendieck gluing are a different specific, but this is a technical matter of finding a more general and flexible setup, like the scheme theory is a natural extension of variety theory. On the other hand, your claim about unifying analytic and algebraic geometry confuses me a bit. I mean it is not only to define the objects – the crucial thing is to control some finiteness conditions. Now it is like approximating a sphere with polyhedra – for algebraic geometry the good objects and morphisms are finitely presented, proper and so on. The best objects in analytic world are far from satisfying such finiteness conditions. This is observed for example in the difference between classical Arakelov geometry and Durov’s algebraic version in his thesis.
However, we show that our models $X/\widehat{Spec \mathbf{Z}}$ (Z. Š.: typography – widehat should be there) of algebraic varieties $X/\mathbf{Q}$ define both a model $\mathcal{X} / Spec \mathbf{Z}$ in the usual sense and a (possibly singular) Banach (co)metric on (the smooth locus of) the complex analytic variety $X(\mathbf{C})$. This metric cannot be chosen arbitrarily; however, some classical metrics like the Fubini–Study metric on $\mathbf{P}^n$ do arise in this way. It is interesting to note that “good” models from the algebraic point of view (e.g. finitely presented) usually give rise to not very nice metrics on $X(\mathbf{C})$, and conversely, nice smooth metrics like Fubini–Study correspond to models with “bad” algebraic properties (e.g. not finitely presented).
Soibelman had some enthusiasm in developing a noncommutative version of Berkovich spectra and was added at the beginning with collaborating Alexander Rosenberg’s spectral experience (and formal methods in descent theory, likewise) in this project and one preprint appeared:
Rosenberg speculated (personal communication) that the existence of Berkovich construction with all its properties in commutative case somewhat accidental from the point of view of a more general spectral theory he studied. Though some examples come from homological mirror symmetry, another impeding problem with nc Berkovich spectra is the lack of guiding examples.
Edit: I put some references at rigid analytic geometry.
14/Durov: that is what I was saying: Finitely presented things approximate all things (the latter being ind-objects, like sphere approximated by a sequence of polyhedra).
I am not sure that I would accept the argument in second part as containing a scheme theory, by just having rings. The basic thing of scheme theory is the study of properties of morphism and how to get to there it is not clear to me from your description. For example, the basic thing how and when do you know that a given morphism in your geometry induces one in algebraic world (specially in nonaffine and relative case); and again questions of finiteness of induced morphisms. In rigid geometry there is a usual thing about the fiber relation with formal schemes. How is this in global geometry ?
Very interesting. Very lively field ! I hope more of it gets into $n$Lab.
It is a bit puzzling to me that schemes are there fully faithfully, I mean faithfully that is OK, but fully…You know one expects to have more analytic morphisms than algebraic ones, unless in some special circumstances, like the projective case.
Also for relative schemes one may need e.g. sheaves over the site of affine schemes over a base scheme (base not necessarily), not just for algebras above a ring (or above integers).
I am late to this discussion here (by about half a year :-) but anyway, here is a comment in reply to #17, where Frédéric said:
Actually, i would prefer a functor of point approach to global analytic geometry (locally representable sheaves on affinoid subspaces of the affine line, the topology being well chosen and also the notion of affinoid…), exactly as Grothendieck’s for schemes.
I would like to have this, too! Moreover, I would like to have the site consisting of affinoids which are contractible – or rather, what I really want is that the site is oo-cohesive.
Of course, I can simply define the site this way. Therefore, what I would really like to understand is how “good” the topos over such a site is, meaning, how faithfully (global) analytic geometry is embeddable in the topos over a site of contractible affinoids.
Berkovich’s result that everly “locally smooth” $p$-adic analytic space is locally contractible suggests that such a site of contractible affinoids would be “good enough”.
Any help in making this more precise would be very much appreciated.
I am sure that many do via functor of points approach (not Berkovich himself). Ayoub has that approach but he looks predominantly on the analytic Nisnevitch site for the obvious reasons. How about Kedlaya’s work ?
P.S. Brian Conrad’s work UNIVERSAL PROPERTY OF NON-ARCHIMEDEAN ANALYTIFICATION pdf is predominantly in the setup of ringed topoi.
P.S.2 maybe of some use – Prop. 2.4 in pdf
Thanks for the links.
Let me maybe say what I think needs to be checked and done.
According to fact 3.4.5. here, every locally smooth analytic space has a cover by contractible ind-objects of affinoids.
So, for what I have in mind, it seems we want to be looking at the site $(Ind Aff)_{contr}$ of contractible ind-objects of affinoids and check, first of all, whether it is a locally infinity-connected site.
For this we need to check if every split hypercover of a contractible ind-affinoid by contractible ind-affinoids has the property that after contracting each component space to a point, the resulting simplicial set is weakly contractible (hence: whether every contractible ind-affinoid is also contractible in étale homotopy.)
First of all, by the above fact we should have covers of any contractible ind-affinoid by contractible ind-affinoids. To promote this to a split hypercover, we need to iteratively cover the intersections of this by contractible ind-affinoids again.
So first of all: are locally smooth analytic spaces stable under intersection/pullback?
I know my last suggestion to pose an MO question led to no answers, but wouldn’t this be a good question to ask there?
I will eventually.
Okay, I have posted something to MO.
Zoran,
concerning the terminology of “rigid” at analytic space:
when looking at any review of the matter, I am getting away with the same impression that Frédéric seems to have expressed above: Berkovich-style analytic geometry is usually presented as going beyond Tate’s rigid analytic geometry. It seems to be misleading to me to speak of Berkovich spaces as “rigid analytic spaces”, since the main point that every review emphasizes is that they are different.
Generally: where does the term “rigid” here come from, anyway? Which aspect of Tate’s spaces is it supposed to reflect? What is rigid about these spaces?
Kedlaya on p. 3 of these slides asks and answers Why Rigid? And he contrasts with Berkovich’s approach.
Yes, that’s how I understand it:
rigid analytic geometry has spaces glued from maximal spectra;
(Berkovich style) analytic geometry has spaces glued from analytic spectra.
Urs, re your MO question. I wonder if you could have posed the question in such a way that analytic geometers knowing nothing about $\infty$-sheaves over $\infty$-cohesive sites could answer.
Right, I’ll try again later.
On the other hand, it is true that some peope say “Berkovich’s theory of rigid analytic spaces”, for instance in the first sentences here.
re #26, on the meaning of “rigid”:
in _p-Adic differential equations on p. 18 Kedlaya explains:
The traditional method is Tate’s theory of rigid analytic spaces, so-called because one develops everything “rigidly” by imitating the theory of schemes in algebraic geometry, but using rings of convergent power series instead of polynomials.
I have added that to the entry rigid analytic geometry.
(And to share a personal opinion: I am glad to know now what “rigid” is supposed to allude to, but I think this is bad choice of terminology.)
Hi Frédéric,
thanks for your reply! This sounds most interesting.
I have the two obvious questions:
could you point me to some references that would help me understand what you mean when you say “…by Kedlaya’s devissage, only when one works with Frobenius modules (Christol-Mebkhout).” ?
is your work that you allude to top secret or would there be a chance to talk with you about it (here or maybe by private email)?
Ah, I guess the Christol-Mebkhout theorem in question is that discussed here.
I have found this here on étale homotopy of analytic spaces, but I don’t see yet that/if it helps with my question above:
Okay, thanks a lot for the references! I am way behind, but maybe I can catch up a little.
Just so that we don’t talk past each other, can you say what general statement it is that you have in mind when you say “I am working hard on this”? Is “this” related to constructing a site of analytic spaces contractible in étale homotopy?
I wonder what tends to happen with ’top secret’ work, such as this and this.
I agree with your attitude here.
Just for the record, on the two specific examples that you link to:
the first one referred to stuff that Alejandro Cabrera had told me about his upcoming thesis, back then. This has long appeared since then. I had seen it once, but right now I can’t find the link… (?)
The second refers to an insight that has meanwhile been fully clarified in section 4.2 of Topological Quantum Field Theories from Compact Lie Groups.
Thanks again, Frédéric. I need to be concentrating on other things now anyway, so I’ll be happy to wait for what you will publish later.
Speaking of publications, you are probably aware of this:
Urs 26: Surely Berkovich introduced new formalism into the wider subject of rigid analytic geometry, not contained in Tate’s Huber did introduce an alternative formalism which has some advantages etc. There are many directions, and it is standard to call them all rigid analytic geometry and all of them feature some variant of rigid analytic spaces. The rigid aspect is seen already at the level of algebraic geometry, let me mention standard phrases like, rigid spaces as the fiber for formal geometry, rigid GAFA, and rigid cohomology…
If one wants to take strictly Berkovich spectrum or Berkovich analytic space and not say, Huber’s version one can call it by that name. In 12 I mentioned some problems with noncommutative version.
let me mention standard phrases like, rigid spaces as the fiber for formal geometry, rigid GAFA, and rigid cohomology…
Yes, and all these are made more flexible by Berkovich spaces. A Berkovich analytic space is not necessarily the fiber of a formal geometry. So it seems misleading to call it by the same name.
In any case, the majority of publications that I have seen distinguishes Berkovich spaces from rigid spaces terminologically, so the entries should reflect that. We can add the emphasis that Berkovich spaces are “just a generalization” as extra information.
I was not aware of publications on the nlab.
It’s only just getting started. So far there is one publication, Leinster2011 (publications), and one more submission is currently still with the referee. After a second publication is through, we are planning to try to formally instantiate the editorial board and then eventually try to establish a recognized journal.
I see in section 4.4 ’Derived Complex Analytic Geometry’ of Structured Spaces Lurie says
With some effort, the ideas presented in this section can be carried over to the setting of rigid analytic geometry. We will return to this subject in [46].
[46] Toric varieties, elliptic cohomology at infinity, and loop group representations. In preparation.
I see Frédéric’s habilitation memoire appeared.
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