Questions tagged [rigid-analytic-spaces]
The rigid-analytic-spaces tag has no usage guidance.
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Reference request: Integral rigid spaces
Is there a notion of integral rigid spaces which mirrors the theory of integral schemes? For instance, let $B$ be an integral affinoid algebra over a non-Archimedean field $k$. Then are the affinoid ...
- 21
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Rational functions on an affinoid subset of $\mathbb{P}^1$
In Van der Put's Rigid analytic geometry and its applications, he's trying to define holomorphic functions on an affinoid subset of $\mathbb{P}^1/K$ as follows, where $(K, |\cdot|)$ is an ...
- 129
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Smooth affinoid rigid-analytic spaces over $\mathbb C_{p}$
Are there any examples of smooth affinoid rigid-analytic spaces $X$ over $\mathbb C_{p}$, which are not the base-change $X=Y_{\mathbb C_{p}}$
of a smooth rigid-analytic variety $Y$ over a finite ...
- 209
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Strong G-Topology on affinoid varieties
I have a question about the strong G-topology of an affinoid variety. In particular let $X=Sp(A)$ be an affinoid variety and we consider analytic functions $f_1,...,f_m\in A$ on X, having no commom ...
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When is the zero set of a multivariate $p$-adic power series algebraic?
Let $f = f(z_1, \dots, z_n)$ be a power series in $n$ variables with coefficients in the $p$-adic integers $\mathbb{Z}_p$. Let $g(z_1, \dots, z_n) = f(pz_1, \dots, pz_n)$, so that $g$ converges on all ...
- 2,677
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Maximal ideals in Tate algebra $K\left<t\right>$
Let $K$ be an algebraically closed nonarchimedean field, $\mathbb{B}=\{ a\in K: |a|\leq 1 \}$ be the unit ball, and let $\text{Sp }K\left<t\right>$ be the maximal spectrum of the Tate algebra $K\...
- 103
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Analytic points in continuous valuation spectra of Huber rings
I am trying to understand a theorem on Huber rings and adic spectra. The specific questions is related to a set of lecture notes by Brian Conrad. The overall question, however, is a fact from one of ...
- 2,188
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Connected components of rational subdomains of rigid line
Let $K$ be a complete discrete valued field, and $f \in \mathcal{O}_K[X]$ be a monoic irreducible polynomial. Let $L$ be the splitting field of $F$, $a \in \mathcal{O}_K$ with $|a| < 1$. Put
\begin{...
- 519
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Coinvariants of rigid meromorphic function (on the p-adic upper half-plane)
Maybe someone can help me out. I am considering the p-adic upper half-plane $\mathcal{H}_p$ given on points by $\mathbb{P}^1(\mathbb{C}_p)\setminus \mathbb{P}^1(\mathbb{Q}_p)$, viewed as a rigid ...
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The base field of the Berkovich unit disk
When defining the Berkovich unit disk $D_k$, we always start with the case that the base field $k$ is algebraically closed. This is because when k is not algebraically closed, we have the isomorphism $...
- 308
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Uniqueness of Weierstrass division for Tate algebras
This question arises from the proof of theorem 8 (Weierstrass division) in chapter 2 of Siegfried Bosch's Lectures on formal and rigid geometry. Given the Tate algebra $T_n =K \langle \zeta_1, \dots, \...
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Rational subsets in rigid analytic and adic spaces
I'm confused about rational subsets of rigid analytic spaces and their corresponding adic spaces.
Suppose $A$ is an affinoid $K$-algebra, $X = \operatorname{Sp}A$ and $f \in A$.
Then $X$ can be ...
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Definition of $p$-adic formal scheme
Could someone please provide a precise definition of a $p$-adic formal scheme $X$ over a ring $A$? Is it a formal scheme over $A$ which is locally isomorphic to $\operatorname{Spf}(B)$, where the ...
- 849
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overconvergence and adic spaces
I have a question on a detail from Huber's Etale Cohomology of Rigid Analytic Varieties and Adic Spaces regarding the equivalence between overconvergent sheaves and sheaves on the partially proper ...
- 21
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Analytic functions on spaces over non-Archimedean fields and troubles with totally disconnectedness
I read in several intro scripts on Berkovish spaces that these
arose as new approach to analytic geometry
over non-archimedean fields. As the main problem in
non-archimedean analytic geometry is ...
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525
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Prerequisites for Rigid Geometry
I'm very interested in learning modern Rigid Geometry, but I'm not sure about the prerequisites for learning it.
I am a 3rd undergrad student majoring in Algebra and Topology. According to you, what ...
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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....
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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 ...
- 359
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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 $\Bbb Q_p[T]$ ...
- 9,338
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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{...
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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^...
- 783
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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 ...
- 362
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69
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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'...
- 105
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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$-...
- 31
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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 $...
- 2,145
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"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 ...
user705174
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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 ...
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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-...
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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+...
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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)$ ...
- 1,273
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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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$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]]$$
...
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Nonarchimedean convergent power series
I would like to understand the second paragraph on the second page (marked page 320) of the article http://www.numdam.org/article/MSMF_1974__39-40__319_0.pdf on rigid analytic geometry by Michel ...
- 680
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(Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$
I am following Scholze's and Weinstein's notes on $p$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates).
$\...
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What are Robba rings and why are they important?
If we study $p$-adic Galois representations we come across the so called "Robba rings", defined as the ring of Laurent series which converge in some annulus $\{z| \,\,R<|z|<1\}$ for some ...
- 1,878
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From sheaf torsors to geometric bundles on schemes
$\DeclareMathOperator{\Spec}{Spec}$
$\DeclareMathOperator{\Sym}{Sym}$
$\newcommand{\func}{\mathcal{O}}$
$\newcommand{\M}{\mathcal{M}}$
It is well known that the notions of locally free $\func_X$-...
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How to determine the minimum number of edges to form a rigid structure in ${\mathbb R}^d$ ($d = 2,3$)?
Let $x_1,\ldots,x_n$ be $n$ points in ${\mathbb R}^d$ ($d = 2,3$). Then, what is the minimum number of edges to form a rigid structure?
Thanks!
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What is a symplectic form of the rotation group SO(n)
I need to prove that the Hamiltonian system of the rigid body motion
$$
\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,
\...
- 159
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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 $g \in \mathcal{A}$,...
- 1,243
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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 $...
- 1,829
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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 ...
- 837
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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 ...
- 6,565
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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 ...
- 1,555
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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/...
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