Questions tagged [rigid-analytic-geometry]

rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields

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When is a coherent sheaf on an algebraizable space algebraizable?

Let $\mathcal{X}$ denote an algebraizable rigid analytic space over $\text{Spm}(\mathbb{Q}_p)$, i.e. there exists a finite type $\mathbb{Q}_p$-scheme $X$ such that $\mathcal{X}$ is its rigid ...
kindasorta's user avatar
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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 $\mathscr{O}_K$, ...
zzp's user avatar
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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 $...
Richard's user avatar
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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 ...
Jef's user avatar
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Kernel of a map of Tate algebras

Let $A$ and $B$ be a pair of Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow A$ must ...
kindasorta's user avatar
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Bezout-type theorem for $p$-adic analytic plane curves

Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
Ashvin Swaminathan's user avatar
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Approximating $p$-adic power series by polynomials

Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
Ashvin Swaminathan's user avatar
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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) ...
Arun Soor's user avatar
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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}}{\...
Richard's user avatar
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Is $\mathbf{C}_p(X)$ self-dual?

Let $X$ be a set. Consider $\mathbf{Q}_p$ and $\mathbf{Z}_p$ as the $p$-adic numbers and $p$-adic integers, respectively. For any finite subset $F \subseteq X$, one can construct the topological ...
Luiz Felipe Garcia's user avatar
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Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita

Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
Vik78's user avatar
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Wondering if Monsky-Washnitzer ever published a result claimed to be forthcoming in a later paper

At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following: "In some sense, the operator $\psi$ applied to a power series gives it "better growth ...
Vik78's user avatar
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Theorem 7.11 in Scholze's $p$-adic Hodge Theory

I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below: Let $...
Kush Singhal's user avatar
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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, ...
James's user avatar
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Notion of connected components for $\mathbb{Q}_p$-points of algebraic variety

Is there an interesting notion of connected components for the $\mathbb{Q}_p$-points of an algebraic variety over $\mathbb{Q}_p$? By "interesting" I mean a notion satisfying the following. ...
Jacques's user avatar
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Adic generic fiber of a small formal scheme in the sense of Faltings

$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
user514790's user avatar
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1 answer
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Restrictions of affinoid Functions from wide open neighbourhoods

Let $X=\operatorname{Sp}(A)$ be an affinoid $K$-space, where $K$ is a p-adic field. Suppose that $X$ lies in the interior of another affinoid $K$-space $X'=\operatorname{Sp}(B)$. Recall that this ...
Tom Adams's user avatar
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G-topological spaces and locales

Consider the following generalization of topological spaces: Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
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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 ...
kindasorta's user avatar
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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) = \{...
Tom Adams's user avatar
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462 views

What lies between algebraic geometry and analytic geometry?

Algebraic geometry and analytic geometry are closely related (witness GAGA). But the latter still seems much "bigger" than the former. I'd like to be able to get from algebraic geometry to ...
Tim Campion's user avatar
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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-...
Sam's user avatar
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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 ...
AZZOUZ Tinhinane Amina's user avatar
3 votes
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209 views

The closed unit adic disk

I am reading the Scholze-Weinstein Berkeley lecture notes on "Perfectoid Spaces", and in particular I am stuck trying to understand the closed adic unit disk, which is the second example of ...
kindasorta's user avatar
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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 ...
Satan's Minion's user avatar
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1 answer
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"Non-algebraic" Berkovich spaces

Usually, Berkovich analytic spaces are derived from some Banach rings (or chains of Banach rings) over a completely normed field $k$ through Berkovich spectrum. But when the base field is the complex ...
Zerox's user avatar
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What's the relation between pseudo-compact and admissible rings?

We recall two definitions. Let $A$ be a linearly topologized ring which is complete and Hausdorff. We say that $A$ is pseudo-compact if, for every open ideal $I\subset A$, the ring $A/I$ is artinian. ...
Gabriel's user avatar
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Can we see the completion of a scheme along a subscheme as an adic space?

Classically, formal schemes were invented to study completions of schemes along closed subschemes. Eventually, people started using them for more arithmetical reasons. (I.e., to study non-archimedean ...
Gabriel's user avatar
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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 ...
kindasorta's user avatar
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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 $\...
kindasorta's user avatar
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On inverse limits of $\pi$-adically complete algebras

Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...
Fernando Peña Vázquez's user avatar
4 votes
1 answer
482 views

On the local properties of rigid analytic varieties

Let $K$ be a non-archimedean field complete with respect to a discrete valuation with ring of integers $\mathcal{R}$, uniformizer $\pi$ and residue field $k$. Consider an affinoid analytic $K$-variety ...
Fernando Peña Vázquez's user avatar
18 votes
1 answer
2k views

Why are there three kinds of non-archimedean geometry?

It may seem silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to ...
Marsault Chabat's user avatar
3 votes
1 answer
162 views

On the stability of having a normal formal model under finite extensions of the base field

Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine ...
Fernando Peña Vázquez's user avatar
8 votes
1 answer
310 views

On actions of finite groups on adic spaces

Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
Fernando Peña Vázquez's user avatar
2 votes
0 answers
115 views

Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$

Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...
KKD's user avatar
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5 votes
1 answer
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On the noetherianess of some subalgebras of an affinoid algebra

$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
Fernando Peña Vázquez's user avatar
2 votes
1 answer
199 views

Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)

Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering $$ \dotsb \...
KKD's user avatar
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8 votes
1 answer
390 views

Noetherian but not strongly Noetherian

What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but not strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \...
Dat Minh Ha's user avatar
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4 votes
0 answers
137 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 $...
Takagi Benseki's user avatar
3 votes
1 answer
141 views

Bounded torsion of quotients of affine formal models

$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
Fernando Peña Vázquez's user avatar
4 votes
1 answer
348 views

On a consequence of the Gerritzen-Grauert Theorem

Let $K$ be a local field of characteristic zero and $X$ an affinoid rigid space over $K$. Let $U\subset X$ be an affinoid subdomain, and consider a finite family of points $\{p_{1},\cdots, p_{n}\}\...
Fernando Peña Vázquez's user avatar
3 votes
1 answer
320 views

Overconvergent modular forms and the level at $p$

I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot. The ...
babu_babu's user avatar
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2 votes
0 answers
157 views

Theorem on formal functions when the initial data is a proper map of formal schemes

Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$. Set $S_0=\{x\}$ be a closed point of $S$ and $...
Adel BETINA's user avatar
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5 votes
1 answer
161 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| = ...
dejavu's user avatar
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4 votes
1 answer
186 views

Higher direct image of coherent sheaf and rigid analytification

Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$ be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) ...
KKD's user avatar
  • 463
3 votes
1 answer
248 views

On the exactness of some completed tensor products

Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...
Fernando Peña Vázquez's user avatar
2 votes
0 answers
211 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 ...
user avatar
1 vote
1 answer
177 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 $k$-...
Dcoles's user avatar
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3 votes
0 answers
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Reference for a p-adic analytic Douady space

I am almost sure that some paper was published in German probably in the 60's or in the 70's proving the existence of a "p-adic analytic Hilbert scheme" (or Douady space) related to a given ...
Antoine Ducros's user avatar

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