Questions tagged [rigid-analytic-spaces]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
1answer
27 views

Given examples of complete non-Archimedean field of characteristic $p$ s.t. $K$ is not $F$-finite

Let $K$ be a complete non-Archimedean field of characteristic $p$ with valuation ring $A$. We say a ring $R$ is $F$-finite if it is of characteristic $p$ and the absolute Frobenius map $F_R$ is finite....
1
vote
1answer
62 views

Restricted power series and tensor product

Let $R$ be a complete adic ring (not necessarily Noetherian) with an ideal of definition $I$. Let $R\left<\zeta\right> := \underset{\longleftarrow}{\lim}_n R/I^n \left[\zeta\right]$ be the ring ...
1
vote
1answer
37 views

Power bounded elements of $\Bbb Q_p \langle T \rangle /(T^2)$

I am rather confused by this example in 9, page 63. Context. Let $A= \Bbb Q_p \langle T \rangle /(T^2)$. $\Bbb Q_p\langle T \rangle$, which as pointed out, this is the completion of $\Bbb Q_p[T]$ ...
-2
votes
1answer
169 views

Does $\varprojlim_{t\mapsto t^p}\mathbb{C}=\mathbb{C}$?

Consider the following inverse system (edit: say category of multiplicative monoids or sets) \begin{equation} \cdots\rightarrow\mathbb{C}\xrightarrow{t\mapsto t^p}\mathbb{C}\rightarrow\cdots \end{...
1
vote
0answers
47 views

Frobenius for function algebras and tilting functor

Consider for example a non archimedean field $K$, then one can take: $$ R = K\langle T^{\pm \frac{1}{p^\infty}}\rangle $$ so in practice if I'm not wrong we' re taking: $$ R \cong \varinjlim_{n}(K[ T^...
2
votes
0answers
180 views

Obtain admissble cover from cover by admissble opens on rigid-analytic spaces

Suppose I have a rigid-analytic $K$-space $X$ and a cover $(U_{i})_{i \in I}$ of $X$ consisting of admissible open subsets, such that each $U_{i}$ satisfies a certain property $P$. Is there any ...
0
votes
0answers
39 views

Notation for Strongly Noetherian Tate k-algebra?

Hey! I am confused with some of the notation - not only in this definition, but also a little bit throughout (section 2 of) Peter Scholze's thesis Perfectoid Spaces. In this situation, what are the T'...
3
votes
0answers
37 views

Construct a subspace of a complete normed $K$-vector space with $(y_\mu)_{\mu \in M}$ as its orthonormal basis

I've been reading Bosch's book "Lectures on Formal and Rigid Geometry". In the proof of Theorem 6 on page 26, which I will show below, he claimed that there is a subspace $V'$ of a complete normed $K$-...
1
vote
1answer
27 views

Image of a local map is closed in $m$-adic topology

I'm reading rigid and formal geometry book of Bosch and in the proof of the proposition 4.2.3 it claims that if $f:A\to B$ is a finite local morphism between two local Noetherian rings such that the $...
2
votes
1answer
98 views

“weak” Henselian property

I encountered following thread from MO treating the structure $X(k)$ of $k$-valued points of a separated algebraic space $X$ of finite type over $k$. I have a question about an aspect from Laurent ...
0
votes
0answers
65 views

Rigid analytic projective line is affinoid?

Let $K$ be an algebraically closed field, complete with respect to a nonarchimedean valuation. Chapter 2 of the book "Rigid Analytic Geometry and Its Applications" by Fresnel and Van der Put begins ...
0
votes
1answer
42 views

Induces a Weierstraß domain an injection of affinoid K-algebras?

Let $X=$Sp$A$ be an affinoid $K$-space and $X(f_1,\ldots,f_r)=\{x \in X ; |f_i(x)| \leq 1\}$ for $f_i \in A$ be a Weierstraß domain in $X$. The inclusion induces a canonical morphism of affinoid K-...
2
votes
0answers
29 views

Morphism of rigid spaces

Let $K$ be an non-archimedean field and $A$ be an affinoid $K$-algebra. Is it then true, that for $f_0, \ldots ,f_n \in A$, the map $\phi:$ Sp$A \rightarrow$ $\mathbb{P}_K^{n,rig}, x \mapsto [f_0(x):\...
4
votes
3answers
146 views

A question about the $p$-adic product formula for $\log(1+X)$ and $p$-adic geometry

Fix a prime number $p$. There is a well known product formula for $\log(1+X)$ as a power series in $\mathbb{Q}_p[[X]]$ (by Serre, I think). Namely letting $\Phi_n(X)=(1+X)^{p^n}-1$ and $Q_n(X)=\Phi_{n+...
1
vote
1answer
93 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 $\Omega_K = \mathbb{P}^1(\mathbb{C}_k) \backslash \mathbb{P}^1(K)$ ...
2
votes
1answer
107 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^{...
1
vote
2answers
72 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 $$f(X)=\sum_J a_JX^J\in \mathbb{k}[[X_1,\dots,X_n]]$$ ...
3
votes
1answer
209 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 ...
7
votes
0answers
210 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). $\...
5
votes
1answer
224 views

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
votes
0answers
190 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 $\func_X$-...
1
vote
1answer
47 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!
2
votes
1answer
142 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, \...
0
votes
1answer
72 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 $g \in \mathcal{A}$,...
0
votes
0answers
81 views

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 $...
0
votes
0answers
135 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
votes
0answers
190 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 ...
4
votes
0answers
101 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 ...
2
votes
2answers
91 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/...