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Questions tagged [rigid-analytic-spaces]

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1answer
35 views

Drinfeld upper half plane admissible open

I have a possibly silly question about the Drinfeld upper half plane. It is "well-know" that if $K$ is a complete local field then $\Omega_K = \mathbb{P}^1(\mathbb{C}_k) \backslash \mathbb{P}^1(K)$ ...
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1answer
31 views

Isomorphism of the perfection of two ring

I was working on Exercise 2.0.4 of Bhatt's notes, which are available here. The exercise states: Let $f\colon R\to S$ be a map of char $p$ rings that is surjective with nilpotent kernel. Then $R^{...
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2answers
51 views

$p$-adic power series and its maximum in the unit ball

Let $\mathbb{K}$ be an algebraically closed field with a complete absolute value and denote by $R$ its valuation ring. Consider a power series $$f(X)=\sum_J a_JX^J\in \mathbb{k}[[X_1,\dots,X_n]]$$ ...
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1answer
123 views

Nonarchimedean convergent power series

I would like to understand the second paragraph on the second page (marked page 320) of the article http://www.numdam.org/article/MSMF_1974__39-40__319_0.pdf on rigid analytic geometry by Michel ...
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0answers
160 views

(Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$

I am following Scholze's and Weinstein's notes on $p$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates). $\...
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0answers
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What are Robba rings and why are they important?

If we study $p$-adic Galois representations we come across the so called "Robba rings", defined as the ring of Laurent series which converge in some annulus $\{z| \,\,R<|z|<1\}$ for some ...
3
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0answers
99 views

From sheaf torsors to geometric bundles on schemes

$\DeclareMathOperator{\Spec}{Spec}$ $\DeclareMathOperator{\Sym}{Sym}$ $\newcommand{\func}{\mathcal{O}}$ $\newcommand{\M}{\mathcal{M}}$ It is well known that the notions of locally free $\func_X$-...
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1answer
43 views

How to determine the minimum number of edges to form a rigid structure in ${\mathbb R}^d$ ($d = 2,3$)?

Let $x_1,\ldots,x_n$ be $n$ points in ${\mathbb R}^d$ ($d = 2,3$). Then, what is the minimum number of edges to form a rigid structure? Thanks!
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1answer
78 views

What is a symplectic form of the rotation group SO(n)

I need to prove that the Hamiltonian system of the rigid body motion $$ \begin{cases} \dot{R}_t=P_tJ^{-1},\\ \dot{P}_t=2R_t\Lambda,\quad\text{$\Lambda$ is the Lagrange multiplier}\\ R_t^T R_t-I=0, \...
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1answer
61 views

Laurent domains are affinoid domains

Let $\mathcal{A}$ be a $k$-affinoid algebra, as in Berkovich (1990, Chapter 2), and let $X := \mathcal{M}(\mathcal{A})$. In (loc. cit. 2.2.2) it is claimed that if $q > 0$, and $g \in \mathcal{A}$,...
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0answers
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Local ring of an affinoid K-space $Spec A$ is not $A_m$

Let $X$ be an affinoid K-space. That is $X=Spec A$ where $A$ is a quotient of the Tate algebra. In Bosch 'lectures on Formal and Rigid Geometry' we find the curious statement that the canonical map $...
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0answers
103 views

Lifting points via étale morphism of adic spaces

This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ...
6
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0answers
143 views

Structure Sheaves of Rigid Analytic Spaces: What is the right Value Category?

Let $K$ be a complete non-archimedean field and let $X$ be an affinoid $K$-space. Then the structure sheaf $\mathcal O_X$ of $X$ is defined first on the "weak Grothendieck topology", ie on affinoid ...
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0answers
80 views

What is a ring $R$ s.t. $Spa(\mathbb{Z},\mathbb{Z})=Spec R$

The adic spectrum $Spa(\mathbb{Z},\mathbb{Z})$ looks as follows: First, there is a point for every prime ideal $\mathfrak{p}\subset\mathbb{Z}$, corresponding to the valuation given by the composition ...
1
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2answers
55 views

Why a finite set is bounded in a topological ring?

I'm almost sure that this question is really idiot and I'm missing something very trivial. In page 36 of Wedhorn's "Adic Spaces" (https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/wedhorn/...