Questions tagged [rigid-analytic-geometry]

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

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9
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0answers
202 views

Why do Coleman functions form a sheaf?

In section 4 of Ammon Besser's 2002 'Coleman Integration Using the Tannakian Formalism,' he defines abstract Coleman functions, which we can describe roughly as those functions which arise by iterated ...
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0answers
152 views

Metrics on vector bundles over Berkovich analytic spaces

$\newcommand{\an}{{\mathrm{an}}} \newcommand{\blank}{{-}} \newcommand{\sheaf}{\mathcal} \newcommand{\comp}{\widehat} \newcommand{\from}{\colon} \newcommand{\IR}{\mathbb{R}} \newcommand{\Spec}{\mathrm{...
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238 views

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 ...
11
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1answer
303 views

Open immersion of affinoid adic spaces

If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of ...
6
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2answers
749 views

Vector bundles on adic spaces

Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? ...
5
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0answers
166 views

$F$-isocrystals defined via a lift of a scheme

Let $X$ be a smooth affine scheme over a finite field $k$. Then there exists a smooth affine formal scheme $\mathfrak{X}$ over $W(k)$ with a lift $\sigma$ of the Frobenius. A convergent $F$-isocrystal ...
2
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1answer
124 views

complement of “good reduction” points in p-adic shimura varieties

assume that $X$ is Siegel Shimura variety defined over $\mathbb{Z}_p$, you can take its p-adic formal completion $\mathfrak{X}$,and than take it's adic generic fiber $\mathcal{X}$ and get an adic ...
9
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146 views

Quasi-separated rigid-analytic space without a formal model?

Well, my question is slightly embarrassing. When learning rigid geometry (mostly from Bosch's book) I realized that I don't know the answer to the following basic question. Question. Is there an ...
1
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0answers
129 views

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 ...
3
votes
1answer
175 views

How to show analytification functor commutes with forgetful functor?

Let $k$ be a field complete with respect to a non-trivial non-archimedean absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$. Denote $X\rightsquigarrow X^{...
7
votes
1answer
162 views

Indeterminacy locus of meromorphic maps of rigid analytic spaces

Setup. Let $k$ be an algebraically closed field of characteristic zero. Let $X/k$ be a normal variety, and let $Y/k$ be a proper variety. It is well-known that the indeterminacy locus of a rational ...
8
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1answer
619 views

On the definition of the etale site of an adic space

I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces". First ...
2
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0answers
109 views

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 ...
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58 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[\![X_1,\ldots, ...
4
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0answers
157 views

nearby cycles map for affine formal schemes

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\...
2
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1answer
165 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 $\...
7
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173 views

Etale maps from smooth affinoids to balls

Let $X$ be a smooth affinoid rigid space over a nonarchimedean field $K$. Does $X$ admit an etale map to a $K$-affinoid ball? The answer is almost certainly "no" in general, but I don't know ...
4
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1answer
251 views

Topological and algebraic covering spaces in Berkovich geometry

Let $k$ be a complete, non-archimedean field, and $X$ a Berkovich space over $k$ (as nice as you like, for arguments sake let's say strictly $k$-analytic, good, and geometrically connected). As ...
4
votes
1answer
254 views

Identity theorem in $p$-adic geometry/analysis

If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $\mathbb{Q}_p$ is disconnected. The modern approach to this is,...
6
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0answers
200 views

Rigid analytic varieties vs rigid spaces

In rigid analytic geometry, some sources refer to "rigid spaces", where others refer to "rigid analytic varieties". Do these two terms stand for the same thing, or is there a ...
12
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1answer
380 views

Can a covering space of the $p$-adic disc split over the circle?

Let $D = {\rm Sp}\, \mathbb{C}_p\langle x\rangle$ be the affinoid unit disc over $\mathbb{C}_p$. Is there an example of a connected finite etale cover of $D$ whose restriction to the "unit circle" ${\...
6
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0answers
328 views

Coherent cohomology of the generic fiber of Lubin-Tate space vs. of Lubin-Tate space considered rationally?

I am trying to compare the coherent cohomology of the generic fiber of Lubin-Tate space to the coherent cohomology of Lubin-Tate space considered rationally, and I am going in circles! I would be very ...
2
votes
1answer
154 views

Reduced complete Tate ring which is not uniform?

Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
4
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0answers
84 views

Existence of a “p-adic Mahler measure” or alternatively, the converge of a p-adic sequence

Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence: $$a_n = \frac{1}{p^{n-1}}\...
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0answers
116 views

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 ...
5
votes
1answer
412 views

Schemes over (locally) ringed spaces: working over complex-analytic spaces, rigid-analytic spaces, formal schemes, etc

Monique Hakim developed in her doctoral thesis [1] the theory of relative schemes. These comprise, as a special case, the theory of schemes over (locally) ringed spaces. What makes the study of these ...
22
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1answer
843 views

Motivation for relative schemes: why should one work with schemes over a ringed topos?

Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
3
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0answers
141 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)...
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93 views

Are affinoid algebras over nontrivially valued fields Jacobson?

It is well-known that for any field $k$ with valuation the Tate algebra $k\{T_1,\dots,T_n\}$ is Jacobson (see Bosch-Güntzer-Remmert for nontrivial valuations; for trivial valuations those are just ...
8
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1answer
746 views

Translation between formal geometry and rigid geometry

I'm reading a paper that translates between formal geometry and rigid geometry. In particular, this paper begins with two rigid analytic spaces $A$ and $C$ (each coming from a scheme over $\mathbb{Z}...
5
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0answers
194 views

Is being perfectoid a local property?

In his 2012 CDM proceedings, Peter Scholze mentions the following open question: Let $K$ be a perfectoid field and $(A,A^+)$ a complete affinoid $K$-algebra. Suppose there exists a cover of $X = \...
2
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0answers
168 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-...
8
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0answers
158 views

Looking for basic example in the theory of adic spaces

I have heard that there exist examples of Huber pairs $(A,A^{\circ})$ and $(B,B^+)$ such that $\operatorname{Spa}(B,B^+)$ is a rational open of $\operatorname{Spa}(A,A^{\circ})$ and such that $B^+ \...
2
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0answers
147 views

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 ...
2
votes
1answer
212 views

Čech complex of rigid $K$-space - Closed image of boundary maps

Let $(X,\mathcal{O}_X)$ be a rigid $K$-space with a finite affinoid covering $(U_i)_{i\in I}$ such that any intersection of the $U_i$ is affinoid too. Equipping the direct products with the maximum ...
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0answers
78 views

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$A \rightarrow$ $\...
7
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1answer
232 views

Are maps corresponding to affinoid subdomains flat in the Banach sense?

$\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\abs}[1]{\lvert #1\rvert}\newcommand{\comptensor}{\mathbin{\hat{\otimes}}}$ Let $k$ be a complete non-archimedian field and let $X = \Sp(B)$ be a $k$-affinoid ...
6
votes
1answer
395 views

An example of a morphism of rigid analytic spaces with affinoid base which is proper but does not satisfy $(\dagger)$

Let $k$ be a complete non-archimedean field and let $\varphi \colon X \to Y$ be a morphism of rigid analytic spaces over $k$, where $\newcommand{\Sp}{\operatorname{Sp}}Y = \Sp(B)$ is affinoid. ...
1
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0answers
54 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 ...
5
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0answers
188 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 :...
3
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0answers
197 views

Is the Čech complex of a coherent sheaf on a compact separated rigid analytic space admissible?

$\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}$Let $X$ be a compact, separated rigid $k$-analytic space over some complete non-archimedean field $k$. Then $X$ has a finite affinoid covering ...
4
votes
0answers
137 views

Rigid cohomology with support and dagger spaces

Let $K$ be a $p$-adic field with residue field $k$, and $X$ a variety over $k$. The rigid cohomology of $X$ over $k$ can be described very neatly using Grosse--Kloenne's notion of dagger spaces: embed ...
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0answers
98 views

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 ...
3
votes
0answers
96 views

A proper analytic surface into which every smooth proper analytic curve embeds

Let $k$ be a finite extension of $\mathbb{Q}_p$. Does there exist a proper $k$-analytic surface such that there is a closed immersion into it from any connected smooth proper $k$-analytic curve? The ...
4
votes
1answer
453 views

Do coherent sheaves on rigid analytic spaces form an abelian category?

It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (...
6
votes
2answers
361 views

What is the difference between total integral closure and integral closure?

I was advised here to make this a new question: What is the difference between total integral closure and integral closure (geometrically, in the context of rigid analytic geometry)? I have read in ...
4
votes
1answer
164 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 ...
5
votes
0answers
119 views

Homotopy of rigid analytic spaces

Let $K$ be a complete non-Archimedean valued field (I think the valuation does not have to be discrete). For a paracompact strictly $K$-analytic space, I have seen at least two definitions of ...
1
vote
0answers
60 views

Failure of Tate acyclicity for integral structure sheaves

Let $(A,A^+)$ be a sheafy Tate-Huber pair, and let $X=\operatorname{Spa}(A,A^+)$. It is well-known that $H^i(X,\mathcal{O}_X)=0$ for $i>0$. I assume it is generally not true that $H^i(X,\mathcal{...
3
votes
1answer
319 views

Crystalline comparison for rigid-analytic varieties

Let $k$ be a finite extension of $\mathbb{Q}_p$. In this paper, Scholze proves an analogue of de Rham comparison for proper smooth rigid-analytic varieties over $k$. He also says: ...it should be ...