Questions tagged [rigid-analytic-spaces]

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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 ...
sriram's user avatar
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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 ...
ALMOST_COMPLEX's user avatar
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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 ...
user141099's user avatar
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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 ...
Running_mathematics's user avatar
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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 ...
Ashvin Swaminathan's user avatar
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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\...
user393795's user avatar
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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 ...
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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{...
fyx1123581347's user avatar
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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 ...
Running_mathematics's user avatar
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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 $...
Yijun Yuan's user avatar
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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, \...
Mark Heavey's user avatar
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1 answer
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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 ...
n7kvz's user avatar
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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 ...
Legendre's user avatar
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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 ...
FaisceauxW's user avatar
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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 ...
user267839's user avatar
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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 ...
Hoang Nguyen's user avatar
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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....
Z Wu's user avatar
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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 ...
E. KOW's user avatar
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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 $\Bbb Q_p[T]$ ...
Bryan Shih's user avatar
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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{...
har_b's user avatar
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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^...
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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 ...
n7kvz's user avatar
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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'...
Brandon Battye's user avatar
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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 $K$-...
ivy's user avatar
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1 answer
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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 $...
ali's user avatar
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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 ...
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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 ...
Mickey's user avatar
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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-...
KKD's user avatar
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6 votes
3 answers
270 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+...
xlord's user avatar
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1 answer
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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 $\Omega_K = \mathbb{P}^1(\mathbb{C}_k) \backslash \mathbb{P}^1(K)$ ...
ggg's user avatar
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4 votes
1 answer
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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 $R^{...
Chen's user avatar
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2 answers
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$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 $$f(X)=\sum_J a_JX^J\in \mathbb{k}[[X_1,\dots,X_n]]$$ ...
Walter Simon's user avatar
3 votes
1 answer
395 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 ...
Mickey's user avatar
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8 votes
0 answers
303 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). $\...
dyf's user avatar
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1 answer
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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 ...
Sameer Kulkarni's user avatar
3 votes
0 answers
301 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 $\func_X$-...
Segipp's user avatar
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1 answer
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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!
Ryan's user avatar
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1 answer
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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, \...
HB khaled's user avatar
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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 $g \in \mathcal{A}$,...
Pedro A. Castillejo's user avatar
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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 $...
Lee Wang's user avatar
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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 ...
Simone's user avatar
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6 votes
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221 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 ...
benh's user avatar
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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 ...
jorst's user avatar
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2 votes
2 answers
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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/...
user40276's user avatar
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