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Questions tagged [rigid-analytic-geometry]

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

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Resolution property in rigid analytic geometry

I am not a rigid analytic geometry, so I apologise if the question is trivial, but I can't find an answer anywhere myself. I'm trying to understand in what ways (rigid) analytic geometry compares to ...
Tim's user avatar
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5 votes
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175 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) $ ...
Theodor's user avatar
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Closed image of curves under $p$-adic logarithm, Coleman integrals and Bogomolov

Disclaimer: my knowledge of $p$-adic analysis/geometry is minimal. Consider a smooth, complete curve $C$ of genus $g$ over $\mathbb{C}_{p}$, denote by $J$ its Jacobian and consider the embedding $C\...
Giulio Bresciani's user avatar
2 votes
0 answers
297 views

Roadmap for p-adic geometry

I think some questions asked in similar fashion with this one. I am a master student in mathematics. I have knowledge in algebraic geometry(both in Shafarevich's and Vakil's books), algebraic topology ...
fyokus's user avatar
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6 votes
1 answer
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What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?

Recently Clausen and Scholze have developed a theory of analytic stack to unify different analytic geometries. Actually I do not know many details about it. It says an analytic stack is a sheaf $\...
Yining Chen's user avatar
5 votes
1 answer
232 views

Closed complement of an open immersion of rigid analytic spaces

I am new to rigid analytic spaces (over non-archimedean fields) and I am confused about the notions of closed and open immersions. My question is are these two notions are "complement" of ...
Alexey Do's user avatar
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Looking for a source on Conrad-Gabber's results about spreading out of rigid-analytic families

Brian Conrad and Ofer Gabber have some results that were announced 9 years ago here: https://www.ihes.fr/~abbes/Gabber/OferGabber.pdf and there's a talk by Gabber about them here: https://www.youtube....
Kim's user avatar
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4 votes
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Compactification of rigid-analytic varieties

Is it true that any separated quasi-compact rigid-analytic variety embeds into a proper one? For my purpose, the base field is a $p$-adic number field. I have seen Huber's universal compactification ...
SmileLee's user avatar
4 votes
1 answer
190 views

Irreducible components of rigid varieties

I'm reading IRREDUCIBLE COMPONENTS OF RIGID SPACES (by Conrad). In this paper he defines the irreducible component of a rigid variety $X$ to be reduced image of a connected component of $\tilde X$ (...
Richard's user avatar
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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 $\mathscr{O}_K$, ...
lolipop'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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3 votes
1 answer
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Kernel of a map of Tate algebras

Let $A$ and $B$ be a pair of noetherian 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 ...
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
3 votes
1 answer
167 views

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
1 vote
0 answers
75 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) ...
Arun Soor's user avatar
1 vote
0 answers
80 views

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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5 votes
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127 views

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
4 votes
1 answer
163 views

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
1 vote
1 answer
79 views

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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4 votes
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180 views

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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3 votes
1 answer
354 views

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
2 votes
1 answer
183 views

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
9 votes
1 answer
335 views

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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1 vote
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81 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 ...
kindasorta's user avatar
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1 vote
1 answer
108 views

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
9 votes
0 answers
505 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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1 vote
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180 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-...
Sam's user avatar
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1 vote
1 answer
105 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 ...
AZZOUZ Tinhinane Amina's user avatar
3 votes
0 answers
224 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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0 votes
1 answer
240 views

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
3 votes
1 answer
459 views

"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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5 votes
1 answer
272 views

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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4 votes
0 answers
305 views

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
  • 1,139
0 votes
0 answers
60 views

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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1 vote
0 answers
134 views

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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4 votes
1 answer
226 views

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
497 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
21 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
165 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
326 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
117 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
  • 463
5 votes
1 answer
351 views

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
207 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
  • 463
8 votes
1 answer
412 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
  • 1,472
4 votes
0 answers
140 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
143 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

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