I see Frédéric’s habilitation memoire appeared.
]]>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 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.
]]>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.
]]>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.
]]>Speaking of publications, you are probably aware of this:
]]>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.
]]>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.
]]>It’s a shame so much has to go on secretly. The dream of the Cafe was to have research open to view. I wonder what tends to happen with ’top secret’ work, such as this and this.
]]>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 have found this here on étale homotopy of analytic spaces, but I don’t see yet that/if it helps with my question above:
Ah, I guess the Christol-Mebkhout theorem in question is that discussed here.
]]>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)?
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.)
]]>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.
]]>Right, I’ll try again later.
]]>Urs, re your MO question. I wonder if you could have posed the question in such a way that analytic geometers knowing nothing about -sheaves over -cohesive sites could answer.
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