Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
243 questions
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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
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0
answers
144
views
Computing the ring of power-bounded elements in an affinoid algebra
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $A$ be an affinoid $K$-algebra, i.e. $A$ is isomorphic to a quotient of the Tate algebra $K\left<T_1,\dotsc,T_n\right>$ for some $n$. ...
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0
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156
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Rigid \'etale cohohomology of flag variety minus its rational points e.g $p$-adic Drinfeld half plane
Let $Fl=G/B$ over $\mathbb Q_p$ be the flag variety of a quasi-split reductive group $G$ over $\mathbb Q_p$, then $X=Fl-Fl(Q_p)$ shall exist as a rigid analytic variety over $\mathbb Q_p$, how to ...
- 6,622
2
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0
answers
257
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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
2
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0
answers
208
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What is meant by a "closed point" of a rigid analytic space?
This is perhaps a naive question, but I have seen some authors (such as Mazur, Kisin) use the term of a "closed point" for (classical) rigid analytic spaces. I am slightly confused as to what this ...
- 643
2
votes
1
answer
247
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Quasi-coherent sheaves on affinoid space
From Conrad's notes on rigid geometry:
More specifically, Gabber has given an example of a sheaf of modules $F$ on the closed unit disk $B^1$ such that $F$ is locally a direct limit of coherent ...
user138661
2
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0
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127
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Status of locality of perfectoidness for uniform rings
Let $k$ be a perfectoid field of zero characteristic. Recall that a Tate $k$-algebra is called uniform if the set of power-bounded elements is bounded. Let $(A, A^+)$ be a uniform complete affinoid $k$...
- 257
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0
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115
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Converging sequence of base change
Here is a natural question that I hope will be of interest to some.
Let $\mathbf{F}_p(\!(T)\!)$ be the field of formal Laurent series over $\mathbf{F}_p$. An automorphism of $\mathbf{F}_p(\!(T)\!)$ ...
- 1,017
2
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0
answers
244
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The composition of proper morphisms in rigid geometry
many books and lecture notes say that it is difficult to prove that the composite of two proper morphisms of rigid analytic varieties are also proper. That is, if two morphisms of rigid analytic ...
- 61
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0
answers
148
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Support of Tor over affinoid algebras
Suppose $k$ is a complete nonarchimedian field, $A$ is a $k$-affinoid algebra, and $M$ is a finitely presented $A$-module. Is the set
$\tau(M)= \left\{ x \in \mathrm{Sp}(A)\,\mathrm{with}\,\mathrm{...
- 13.1k
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0
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455
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reference for p-adic Stein spaces
Hi,
I'm looking for a reference in english for p-adic Stein spaces. The usual referneces I come across are all in german.
Thanks
- 2,842
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
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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
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1
answer
242
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Can a p-adic ball cover a p-adic ball?
Are there a polynomials $f_1,...f_n \in \mathbb{Z}_p[x_1,...x_n]$ with there coeficients $p$-adic integers s.t.
A map $F:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p^n$ defined by $f_1,...f_n$
satisfy the ...
- 328
1
vote
1
answer
98
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Geometrically connected affinoid cover of geometrically connected smooth rigid space
I am interested in the following question:
Given a geometrically connected smooth rigid analytic space $X$ over a non-archimedean field $k$, is it always possible to find an affinoid open covering, ...
- 208
1
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1
answer
762
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What are in units of an affinoid algebra?
Suppose that $K$ is a complete local field and $A$ is an affinoid $K$-algebra. Is there a known way to produce an explicit description of the units of $A$?
Here is what I already know: write $A^\circ$...
- 3,545
1
vote
1
answer
131
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Reduction of elliptic curves over local fields
Let $E$ be an elliptic curve defined over a p-adic local field $K$, with $j$-invarient $j(E)\in K$. Let $\mathscr{O}_K$ be the ring of integer of $K$. If $j(E)$ does not belong to $\mathscr{O}_K$, ...
- 95
1
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1
answer
133
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Injectivity of sheaf restriction maps for wide open neighbourhoods of rational subdomains
Let $X=\text{Sp}(A)$ be an affinoid $K$ space, where $K$ is a $p$-adic field. If $f_0, f_1,..., f_s \in A$ generate the unit ideal then we can define the rational subdomain $U= X(f_0, f_1..., f_s) = \{...
- 117
1
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1
answer
209
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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
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0
answers
73
views
Krull dimension of affinoid algebra
Let $K$ be a complete field w.r.t. a valuation, with residue field $k$. Let $A$ be an affinoid algebra over $K$ with respect to a valuation $V$ (in the sense of Tate; in the terminology of Berkovich, $...
- 237
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0
answers
98
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Points on a rigid analytic variety and "points" on a formal model
Let $k$ be a finite extension of $\mathbb{Q}_p$. Let $X$ be a quasi-compact, quasi-separated rigid analytic variety over $k$. We choose a formal model $\mathcal{X}$ of $X$ over $\mathcal{O}_k$.
If I ...
- 519
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0
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86
views
Increasing coverings of rigid analytic varieties
Let $K/\mathbb{Q}_p$ be complete and let $X/K$ be a rigid analytic variety. When does $X$ admit an "increasing" admissible covering by quasi-compact admissible (in the strong G-topology) ...
- 36
1
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0
answers
80
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The bound for zeros of the composition of polynomials and analytic functions
Suppose $K$ is a number field, and $A\in M_n(K)$. $v$ is a place of $K$, and $f_1,\cdots,f_n$ are analytic functions (one variable) on $m_v\mathcal O_{K,v}$, satisfying: $\frac{\mathrm d \bf {f}}{\...
- 785
1
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0
answers
84
views
Algebraizable image of a morphism of Galois cohomology stacks
Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...
- 2,907
1
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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-...
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0
answers
137
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The stack $\operatorname{GL}_2/B$
Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...
- 2,907
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0
answers
190
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Moduli interpretation of normalization of moduli space
The question is about formal and rigid geometry, but I would be interested in an answer from an algebraic geometry point of view as well.
Let $\mathfrak{X}$ be a formal moduli space (e.g., the formal ...
- 949
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0
answers
81
views
Completion of $K$-algebra of finite type with respect to the residue norm
Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\
- 473
1
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0
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177
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L-function in p-adic spaces
I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate ...
- 4,418
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0
answers
87
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Morphism into K-analytic projective space $\mathbb{P}_K^{n,rig}$
Let $K$ be a local field, with a completed algebraic closure $\hat{\bar{K}}$ and $A$ be an affinoid $K$-algebra. Is it then true, that for $f_0, \ldots ,f_n \in A$, the map $\phi:$ Sp$A \rightarrow$ $\...
- 473
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0
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80
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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
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0
answers
209
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Quasi-coherent sheaves in Tate's theorem
A theorem of Tate states that for any affinoid space $X=\mathrm{Sp}\:A$ and a coherent sheaf $F$ we have $H^i(X,F)=0$ for all $i>0$.
Is this theorem true for merely quasi-coherent sheaves? The ...
user143954
1
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0
answers
131
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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
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0
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256
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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
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0
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165
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Rigid analytic reductions of the projective line
I'm reading the book "Rigid Analytic Geometry and its Applications" by Fresnel-van der Put, and I'm confused by their example 4.8.5. In the first two parts of the example, they define the analytic ...
- 164
1
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0
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348
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rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
- 147
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0
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115
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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
1
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0
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522
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Component group of Neron model of a parametrized abelian variety
Let $A$ be an abelian variety of dimension $2$ over a $p$-adic field $K$ with (additive) valuation $v$. Assuming $A$ has multiplicative reduction, the theory of $p$-adic theta functions gives us an ...
- 15.4k
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1
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287
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When is the power-bounded subring top. of finite type?
Very naive question here. Let $K$ be a complete nonarchimedean field, $A$ a reduced affinoid $K$-algebra. When is the power-bounded subring $A^\circ$ topologically of finite type, in the sense that we ...
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0
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111
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Prime to $p$ monodromy of local system on rigid variety
Suppose $F$ is a finite extension of $\mathbb Q_p$, and $X$ is a rigid variety over $F$. I saw in proposition 3.7 of Oswal, Shankar, Zhu, and Patel - A $p$-adic analogue of Borel's theorem: "Let $...
- 785
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0
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64
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Weierstrass subdomain of $\DeclareMathOperator\Spm{Spm}\Spm \mathbb{Q}_p$
I am trying to understand Weierstrass subdomains of $\Spm\DeclareMathOperator\QP{\mathbb{Q}_p}\QP$.
Recall that a Weierstrass algebra of an affinoid space $\Spm A$, where $A$ is a Banach algebra with ...
- 2,907
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416
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Analytic structures on the source of a surjection of condensed rings
Question. Let $(\mathcal B,\mathcal N)$ be an analytic (animated associative) ring, $\mathcal A$ be a condensed (animated associative) ring and $f\colon\mathcal A\to\mathcal B$ a surjective map of ...
- 2,856
0
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0
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318
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morphism from adic spaces to schemes
Let $X:=Spa A$ be an affinoid adic space, and $\underline X $ the ringed space of $X$. Let $Y:=Spec B$ be an affine scheme, $f: \underline X \longrightarrow Y$ a morphism of ringed spaces.
How to ...
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