Questions tagged [rigid-analytic-spaces]
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29
questions
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/...
