All Questions
Tagged with rigid-analytic-geometry non-archimedean-fields
29 questions
45
votes
2
answers
4k
views
Are rigid-analytic spaces obsolete, since adic spaces exist?
Recently in a seminar the following question was raised and, despite my familiarity with theory, I couldn't come up with a good answer:
Are there any good reasons to use Tate's theory of rigid-...
- 28.2k
18
votes
1
answer
1k
views
Why do rigid spaces have "not enough points"?
In Brian Conrad's notes
here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...
- 585
12
votes
1
answer
885
views
bornological vector spaces over a non-archimedean field
Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
- 1,180
5
votes
1
answer
471
views
Rigid analytic geometry in characterstic 0 vs positive characteristic
This question is motivated purely by curiosity. In algebraic geometry there is a major distinction between the world of characteristic $0$ and that of characteristic $p > 0$ with different methods, ...
- 912
5
votes
1
answer
179
views
An example where the non-Archimedean tensor product of normed modules is only seminormed?
Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies
$$ | 0_R| = ...
- 153
5
votes
2
answers
596
views
generic fibre functor for relative rigid spaces
The classical theory of formal models of rigid analytic spaces due to
Raynaud introduces the category of admissible R-formal schemes for $R$ a
discretely valued ring, which includes locally ...
- 4,070
5
votes
1
answer
333
views
Topology of ring of global sections of finite union of affinoid opens in a rigid analytic space
Let $X$ be a rigid analytic space over a non-Archimedean field $k$. If $U_1,\ldots,U_n\subseteq X$ are affinoid opens, then it's usually not clear whether or not the admissible open $U=U_1\cup\cdots\...
- 3,380
5
votes
0
answers
438
views
Stalks of nonarchimedean spaces as analytic rings
Let $(A,A^+)$ be an affinoid Tate ring, and let $x \in X=\operatorname{Spa}(A,A^+)$. When defining the stalks of the structure sheafs ${\mathcal O}_{X,x} = \varinjlim_{x \in U} {\mathcal O}_{X}(U) $ ...
- 151
5
votes
0
answers
405
views
Is this subset of a rigid space an admissible open?
Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball.
Consider the sets $U := \{ |T_1| < 1\}$ and $V :...
- 37k
4
votes
2
answers
622
views
Is there a notion of pure dimension for Berkovich analytic space?
For affinoid spaces the definition is similar to algebraic geometry, what about general analytic spaces? I can't find a reference about it. If yes then is the analytification of a variety of pure ...
- 129
4
votes
1
answer
225
views
Significance of integrally closed in an affinoid algebra
A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.
See for ...
- 7,746
4
votes
1
answer
464
views
Tropical charts (coordinates) and differential forms in non-archimedean geometry
Chambert-Loir and Ducros have introduced real differential forms and currents on Berkovich spaces.(See Gubler's survey for example). In that survey, a tropical chart $V$ is defined on an ...
- 2,789
4
votes
1
answer
469
views
Spherical completions and flatness
Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? ...
- 1,180
4
votes
1
answer
483
views
Tate algebras and fundamental theorem of algebra
Let $\mathbb K$ be an algebraically-closed complete non-archimedean field whose absolute value is non-trivial. Consider the Tate algebra $T_n=\mathbb K\langle X_1,\dots, X_n \rangle$ and fix $f\in T_n$...
- 2,789
4
votes
0
answers
117
views
Projective reduction of image of power series is algebraic?
Let $K$ be a non-archimedean field with closed unit disk $\mathcal{O}\subset K$, open unit disk $\mathfrak{m}\subset \mathcal{O}$ and residue field $k = \mathcal{O}/\mathfrak{m}$.
Examples to keep in ...
- 984
4
votes
0
answers
149
views
Coherence of the I-adic completion of a local ring of a formal scheme
Let $K$ be a valued field of rank one and $K^+$ its valuation ring such that $K^+$ is $\varpi$-adically complete with respect to a pseudo-uniformizer $\varpi\in K^+$. Let $X$ be a smooth finite type $...
- 119
3
votes
1
answer
411
views
Why does $\mathbb C_p$ not contain the periods?
I am reading the following article of Berger, p8 and I don't understand the idea:
$C_p:=\widehat{\overline{\mathbb Q_p}}$ does not contain the periods
The text seem to reason as follows
(under some ...
- 661
3
votes
0
answers
183
views
gluing Berkovich spaces
In his paper Etale cohomology for non-Archimedean analytic space (IHES), Berkovich explained how to glue $k$-analytic spaces (Prop. 1.3.3) and show its uniqueness using the Prop 1.3.2 (gluing morphism)...
2
votes
1
answer
300
views
Is $\mathbb{A}_k^n(k)$ dense in the Berkovich analytification of $\mathbb{A}_k^n$?
Let $k$ be a non-archimedean field and denote by $\mathbb{A}_k^n$ the analytic affine space of $n$ dimensions over $k$ (analytic in the sense of Berkovich). There is a natural injective map of sets $\...
- 1,153
2
votes
0
answers
250
views
Enlightening examples of tropical skeletons of Berkovich spaces
Let $K$ be a complete non-archimedean field and let $X$ be a $K$-analytic space in the sense of Berkovich of pure dimension $d$. Let $\varphi \colon X \to \mathbf{G}_m^r$ be a moment map to an ...
user478030
2
votes
0
answers
257
views
Rigid analytic geometry and Tate curve
I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-...
user141691
1
vote
2
answers
766
views
Berthelot functor, rigid analytic space
If $X=\operatorname{Spec} A$, where $A$ is a noetherien, complete local ring, with a finite residual field $\mathbb{F}_p$. We can associate to $A$ a rigid analytic space with two different ways, we ...
- 1,066
1
vote
1
answer
111
views
Complete residue field of a point of type 5
Let $(F,|.|)$ be a complete algebraically closed field. Let $x$ be the point of type 5 corresponding to the unit open disc of the adic affine line over $F$. Can we obtain a concrete description of the ...
1
vote
1
answer
209
views
Reference request: Gruson's theorem on the tensor product of Banach spaces over a non-Archimedean field
I am looking for a reference for theorem 3.21 of these notes: https://web.math.princeton.edu/~takumim/Berkovich.pdf
The theorem states that if $k$ is a non-Archimedean field and $X$ and $Y$ are $k$-...
- 63
1
vote
0
answers
183
views
Contractibility of the quotient of an analytification of a smooth variety by a finite group (if the field is trivially valued)
Let $k$ be a field and $X$ be a smooth irreducible $k$-variety with an action of a finite group $G$. I consider $k$ as a trivially valued field.
It is known from results of Berkovich ("Smooth p-...
- 41
1
vote
0
answers
80
views
Polytopal domains in non-archimedean torus
Given a non-archimedean field $\mathbb K$, there is a natural map
$$
\mathrm{val}: (\mathbb K^*)^n\to\mathbb R^n$$
(See Section 4 of Gubler's paper).
Gubler mentions there $\mathrm{val}$ is a ...
- 2,789
1
vote
0
answers
131
views
Affinoid algebra and fundamental theorem of algebra
This post is closely related to the previous one here.
But more generally, we want to study an affinoid algebra $A:=T_n/\mathfrak a$. Let's assume $\mathfrak a= (f_1,\dots,f_r)$ for some $f_i\in T_n$....
- 2,789
1
vote
0
answers
256
views
Explicit description of rigid analytification of torus
It is known that in non-archimedean world there is also a GAGA-functor from the category of $K$-schemes of locally finite type to the category of rigid $K$-spaces. Here $K$ is a field with a non-...
- 2,789
1
vote
0
answers
115
views
Relative nonarchimedean disks and annuli
Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition.
Is there a good notion of closed disk of ...
- 3,605