45
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$p$-adic Hodge Theory for rigid spaces, after P. Scholze
Let me start with the second question first:
The usual de Rham complex is not locally acyclic in positive degrees, in any of the topologies (analytic (= of rational subsets), étale, pro-étale, ...). ...
- 21.3k
31
votes
Are rigid-analytic spaces obsolete, since adic spaces exist?
My opinion (knowing that this is somehow a matter of taste) is that the rigid-analytic viewpoint is more or less obsolete, and that one should better use Berkovich, Huber or Raynaud instead (each of ...
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Are rigid-analytic spaces obsolete, since adic spaces exist?
There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications?
For the first question, it'...
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votes
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Why are there three kinds of non-archimedean geometry?
Tate's rigid-analytic geometry was the first theory of (global) nonarchimedean geometry to have been devised, and in some sense could be seen as a "proof of concept" that such a theory can ...
- 28.2k
19
votes
Perfectoid universal covers
Lol @"varying degrees of enthusiasm" ;-). And sorry for the late answer...
Let me try to answer your questions. First, for any connected analytic adic space $X$, say, with a geometric point $...
- 21.3k
16
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simple questions on topological rings arising in the context of Perfectoid Spaces
Firstly, I don't think you should start to learn about adic spaces by thinking about perfectoid spaces. The sensible examples of adic spaces for a beginner to think about are sane Noetherian things ...
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15
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Translation between formal geometry and rigid geometry
No, these are not the same thing. Formal schemes are to rigid-analytic spaces as $\mathbf{Z}_p$-schemes are to $\mathbf{Q}_p$-schemes.
The book Lectures in Formal and Rigid Geometry by Bosch is an ...
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Motivation for relative schemes: why should one work with schemes over a ringed topos?
The comments to your question discuss the variation of relative schemes over a topos, vs relative schemes over a site. But it seems your question
stood at the more basic level of the relevance of ...
- 12.9k
14
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Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)
In a nutshell:
$\mathrm{Spec}~\mathbb{C}[[t]]$ Is a “trait” i.e. the spectrum of a discrete valuation domain, with a generic (open) point and a closed point. It might be imagined as a refinement of ...
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13
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On the definition of the etale site of an adic space
Great question!
The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat ...
- 21.3k
11
votes
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Flatness of the integral closure
No. Take $R=\mathbb{Z}_p[x,y]/(xy-p^2)$, $S^+=\mathbb{Z}_p[u,v]/(uv-p)$, and map $R$ into $S^+$ by $x\mapsto u^2$, $y\mapsto v^2$. Note that $R$ is normal, $S^+$ is a finite extension of $R$, étale of ...
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11
votes
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Finite subgroups of GL_n of polynomial rings over finite fields
Well, polynomial rings are very complicated. I assume the question is more generally about $GL_n(\mathbb{F}_q[T_1,\dots,T_m])$.
The case of one variable can be deduced from the following paper:
C. ...
- 17.4k
11
votes
A roadmap for understanding perfectoid spaces
I can tell from personal experience that it is possible to learn perfectoid spaces without knowing rigid geometry, just like it is possible to learn schemes or even stacks without knowing much about ...
- 127
11
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On actions of finite groups on adic spaces
The issue with your calculation is, to put it bluntly, that fiber products of adic spaces are weird, and more specifically, they cannot be computed naively at the level of points - categorically ...
- 28.2k
10
votes
A roadmap for understanding perfectoid spaces
I agree with nfdc23, notwithstanding here is some kind of 'roadmap' but I am by no means an expert.
I would begin with Brian Conrad's "Several approaches to Non-Archimedean geometry" chapter in a set ...
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10
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A roadmap for understanding perfectoid spaces
Apart from all the written sources, there are six videos of Peter Scholze's lectures at the IHÉS in 2011 on Perfectoid Spaces and the Weight-Monodromy Conjecture :
http://www.dailymotion.com/video/...
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9
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Why does $\mathbb C_p$ not contain the periods?
Consider the Tate motive $\mathbb Q(1)$. Its de Rham realization is simply $\mathbb Q$ (with the filtration $F^{-1}\mathbb Q=\mathbb Q$ and $F^{0}=0$) and its Betti realization is $2\pi i\mathbb Q$. ...
- 10.9k
8
votes
Vector bundles on adic spaces
The question is local on $X$, so we may assume that $\mathcal E$ is finite free, of rank $n$, say. In that case, as also SashaP points out, the question amounts to the question whether $\mathbb A^n_X$ ...
- 21.3k
8
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Vector bundles on the various sites of a preperfectoid
There is some relevant work of Ben Heuer on this.
In short, analytic and etale vector bundles agree on all sousperfectoid adic spaces (and much more generally, for all "etale sheafy" adic ...
- 21.3k
8
votes
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Noetherian but not strongly Noetherian
The only example I know of occurs in On Hausdorff completions of commutative rings in rigid geometry by Fujiwara, Gabber, Kato (and according to the intro the example is due to Gabber). The example is ...
- 466
7
votes
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Reference Request: Specialization map in Huber's Context
In case you're still interested: Bhatt recently proved that for any Tate-Huber pair $(A,A^+)$, the topological space $\mathrm{Spa}(A,A^+)$ is homeomorphic to an inverse limit of admissible blowups in ...
- 13.1k
7
votes
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$p$-adic Kato--Nakayama space
I believe many people have thought about this at some point, and I don't think such a construction is known.
In the paper https://arxiv.org/abs/1207.3380 , Yves Andre considers the real blowups (in ...
- 16.1k
7
votes
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Are maps corresponding to affinoid subdomains flat in the Banach sense?
This is not true. Assume that $X$ is the closed unit disc (given by $|T| \le 1$, with algebra $B$) and $V$ is a smaller disc (given by $|T| \le r$ for some $r \in (0,1)$, with algebra $B_V$). Consider ...
- 4,132
7
votes
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What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?
One way to think about categories of stacks (or more generally, (higher) topoi) is that one has some class of generating objects, and some class of allowed gluings; and then one builds the full ...
- 21.3k
7
votes
Help with understanding a rigid geometry proof
I find this argument a bit difficult to follow (how does he deal with points of multiplicity $>1$?). Here is an alternative proof using formal models.
Since $Y$ is a smooth rigid-analytic curve, ...
- 16.1k
6
votes
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Rigid versus log-rigid cohomology for semistable varieties
$\require{AMScd}$I'll expand a little on my comment to give an answer to David's follow up question:
Firstly, the general relationship is described in Chiarellotto's Duke 1999 paper "Rigid cohomology ...
- 1,404
6
votes
Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)
One good way of seeing the difference between $\mathrm{Spec}\,\mathbb{C}[[t]]$ and $\mathrm{Spf}\,\mathbb{C}[[t]]$ is to look at what functors they represent on affine schemes. In fact we have
$$\...
- 16.5k
6
votes
Accepted
Abel-Jacobi map for Mumford curves analytically
This is done in Manin and Drinfeld's "Periods of p-adic Schottky groups." Journal für die reine und angewandte Mathematik 0262_0263 (1973): 239-247. You will also find it in Gerritzen and van der Put'...
- 4,132
6
votes
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Do coherent sheaves on rigid analytic spaces form an abelian category?
The answer is yes.
An abelian category is a category $\mathcal C$ with the following properties:
$\mathcal C$ is additive.
$\mathcal C$ has kernels and cokernels.
Images and coimages coincide. That ...
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6
votes
Accepted
Can a covering space of the $p$-adic disc split over the circle?
$\newcommand{\Sp}{\mathrm{Sp}\,}\newcommand{\cO}{\mathcal{O}}\newcommand{\fm}{\mathrm{m}}$There seem to exist such examples and this does not contradict de Jong's argument because his proof only shows ...
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