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Questions tagged [rigid-analytic-geometry]

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

5
votes
1answer
218 views

Flatness of the integral closure

Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$. Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...
1
vote
0answers
99 views

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 $k$...
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vote
0answers
87 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 $f_i\in T_n$....
1
vote
0answers
69 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-...
3
votes
1answer
271 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 $f\in T_n$...
3
votes
0answers
113 views

Etale cohomology of projective spaces in the rigid analytic setting

Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...
5
votes
0answers
151 views

Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$

$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...
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0answers
643 views

Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
10
votes
0answers
204 views

Zeros of $p$-adic power series and rationality

Let $K$ be a non-archimedean field with valuation ring $(V,\mathfrak{m})$, and $K\langle t_1,\ldots, t_n\rangle$ a Tate algebra of convergent power series. Fix $f \in V\langle t_1,\ldots, t_n\rangle$....
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0answers
240 views

Why are the open and closed adic discs defined the way that they are?

The closed adic disc is defined as $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$, and the open adic disc is defined to be the fiber $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
9
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0answers
264 views

What role, if any, do Archimedean valuations play in adic spaces?

I've been reading about adic spaces, and I couldn't help but wonder what would happen to the theory if one included in the definition of $Spa$ Archimedean valuations as well...? Is there a weird ...
4
votes
1answer
275 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 ...
4
votes
1answer
220 views

Reference Request: Specialization map in Huber's Context

The specialization map $sp:\mathfrak{X}_\eta\to \mathfrak{X}_{red}$ has an important role in rigid analytic geometry. I tried looking in Huber's papers ("Continuous Valuations", "A generalization of ...
2
votes
1answer
206 views

A translation between formal and rigid geometry

The following lemma is in Bosch's book "Lectures on Formal and rigid geometry" p198. Lemma Let $K$ be a non-archimedean field and $R$ its valuation ring. Let $X= \mathrm{Spf}A$ be an affine ...
10
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0answers
317 views

Symplectic structures in rigid geometry

Let $K$ be a non-archimedean valued field (with any further adjectives attached as necessary). I'm looking for references or information about symplectic structures on rigid $K$-spaces. For example, ...
4
votes
0answers
119 views

Complete characteristic p perfect Tate rings are uniform?

In Lemma 7.1.6 of his lecture notes on perfectoid spaces, Bhatt states that every complete characteristic p perfect Tate ring $A$ is uniform. In the proof he uses the Banach open mapping theorem on ...
10
votes
1answer
425 views

(pro)Étale cohomology of adic spaces and inverse limit

I am studying Peter Scholze's paper $p$-adic Hodge theory for rigid-analytic varieties and I am confused by the following. Let $X$ be a finite type scheme over $\mathbb{C}_p$ (proper and smooth if ...
20
votes
3answers
631 views

Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)

The first one $\mathrm{Spec}\mathbb{C}[[t]]$ is a scheme, the second one $\mathrm{Spf}\mathbb{C}[[t]]$ is a formal scheme. In my mind they both realize an "infinite order infinitesimal neighbourhood ...
5
votes
0answers
429 views

Formal GAGA and étale cohomology

Let $\mathfrak{X}$ be a $p$-adic flat formal scheme over $\mathbf{Z}_p$, whose special fiber has an ample line bundle. Then $\mathfrak{X}$ is algebraizable, that is there exists an algebraic $\mathbf{...
5
votes
0answers
102 views

Ring of functions of generic fiber of affine special formal schemes

Fix $R$ a complete DVR. Recall from Berkovich's Vanishing Cycles for Formal Schemes II paper that we have a class of special formal schemes which are not topologically of finite type over $\...
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vote
0answers
49 views

can one extend a coordinate function of an open annulus to an affinoid domain?

Fix a field $k$ endowed with a non-Archimedean absolute value. Let $A$ be a rigid analytic space isomorphic to an open annulus, i.e. a domain in $\mathbb{A}_k^1$ defined by the inequalities $r < |...
3
votes
0answers
133 views

Analytic Space with No Regular Points

Define an analytic space to be a topological space $X$ equipped with a sheaf of rings $\mathcal{O}_X$ such that for every point $x \in X$ there is a neighbourhood $U \subseteq X$ such that $(U, \...
29
votes
1answer
3k views

$p$-adic Hodge Theory for rigid spaces, after P. Scholze

I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties. This question is around the "Poincaré Lemma" in the paper. Throughout, let $X$ be a proper smooth rigid ...
6
votes
2answers
367 views

$p$-adic exponentials for $p$-adic Lie groups

Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra. Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?
10
votes
3answers
927 views

p-adic Poincaré Lemma

suppose $X$ is a proper and smooth rigid analytic variety over $\text{Spa}(k)$, with $k$ a non-archimedean field of characteristic zero. One has the de Rham complex of analytic differential forms on $...
3
votes
0answers
157 views

Seminorms on tensor products of affinoid algebras

Let $A \leftarrow C \rightarrow B$ be affinoid $K$-algebras, where $K$ is a non-archimedean field with non-trivial absolute value. Equipping $A$, $B$, $C$ with the supremum seminorms, there is a ...
3
votes
0answers
227 views

Etale cohomology of rigidification

Let $K/\mathbb Q_p$ be a discretely valued non-archemedean field, let $X$ be a smooth scheme over $\mathcal O_K$. To $X$ one can associate two rigid-analytic spaces over $K$: 1) the analytification $...
8
votes
0answers
240 views

Disconnectedness in Rigid Analytic Geometry: a technical question

I have a technical question in rigid analytic geometry, in the sense of Tate. Let $B$ be an affinoid space over $\mathbb Q_p$, $Z$ a rigid analytic variety with a flat surjective map $f:Z \...
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vote
0answers
120 views

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 ...
3
votes
1answer
180 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 ...
10
votes
0answers
194 views

Kodaira embedding theorem for rigid analytic varieties

Kodaira embedding theorem can be regarded as a vast generalizaton of the projectivity criterion for complex tori: indeed, the Riemann conditions essentially say that the line bundle defined by the ...
2
votes
1answer
280 views

Representation of elements in affinoid algebra

Let $K$ be a complete, algebraically closed non-Archimedean field, and let $p \in K[x]$ be of degree $d > 0$ and norm 1. (Here the norm of a polynomial is the maximum of the norms of its ...
1
vote
0answers
288 views

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 $\mathbb{F}_q$, $q=p^m$, $...
2
votes
0answers
111 views

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 $\mathbf{F}_p(\!(T)\!)$ ...
6
votes
2answers
222 views

Abel-Jacobi map for Mumford curves analytically

Let $K$ be a field equipped with a non-Archimedean absolute value, let $\Gamma$ be a Schottky group in $PGL_2(K)$, and let $X_\Gamma$ be the associated Mumford curve, which is a proper smooth rigid ...
3
votes
0answers
253 views

Cyclotomic Extension of a Perfectoid Space

Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...
3
votes
0answers
186 views

Compact subgroups of general linear groups over affinoid algebras

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $(A,A^0)$ be a $k$-affinoid algebra, where $A^0$ is the subring of power bounded elements. Suppose given a compact subgroup, L, of $GL_n(A)$, is ...
4
votes
1answer
681 views

simple questions on topological rings arising in the context of Perfectoid Spaces

(I apologize in advance for these simple questions, I am a beginner trying to go through Scholze's paper Perfectoid Spaces). Let $(R, R^+)$ be an affinoid $k$-algebra as defined in Scholze's paper ...
18
votes
3answers
4k views

A roadmap for understanding perfectoid spaces

Perfectoid spaces are this year's subject for the Arizona Winter School (link) and, as preparation, I am currently trying to understand the subject better. There are wonderful explanatory accounts (...
5
votes
1answer
276 views

Finite subgroups of GL_n of polynomial rings over finite fields

I am surprised that I didn't find a reference for the following question. Q: Is there any characterization of the finite subgroups of $GL_n( \mathbb{F}_p [T_1, \dots, T_n])$? Can we do so more ...
2
votes
0answers
188 views

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 ...
2
votes
0answers
181 views

Representability of relative Hilbert and Picard functors over analytic spaces

Let $f:X \to S$ be a morphism of complex analytic spaces. Then, just like in the case of schemes, we can define the relative Hilbert and Picard functors. For instance, if $\text{An}_{/S}$ denotes de ...
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votes
0answers
91 views

Group action of $ GL_{2}(\mathbb{Q}_p) $ on Bruhat-Tits tree

When I studied a p-adic version of the Beilinson-Bernstein localisation theorem for representations (cf. "LOCALLY ANALYTIC REPRESENTATIONS OF GL(2,L), VIA SEMISTABLE MODELS OF P1" by D. Patel, T. ...
33
votes
1answer
1k views

Perfectoid universal covers

It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this ...
5
votes
1answer
382 views

Schottky groups, Mumford curves and $p$-adic uniformization

Let $K$ be a $p$-adic field and $\Gamma$ be Schottky group of $g$ generators and $L \subset \mathbb{P}^1_K$ be the limit set of $\Gamma$. Let $\mathbb{P}^1_K - L= \Omega$ and we know that there exists ...
4
votes
0answers
421 views

Compatibility of formal completion and rigid analytic generic fiber

Let $R$ be a complete valuation ring of rank $1$ (e.g., a complete discrete valuation ring) and let $K$ be its field of fractions. Consider a proper $R$-scheme $X$ that is, say, normal (if needed). ...
8
votes
1answer
293 views

Good analytic spaces over a field into locally ringed spaces is fully faithful

Let $k$ be a field which is complete with respect to a non-trivial non-archimedean rank-1 valuation, and let $X$ be scheme which is locally of finite type over $k$. In section of 3.5 of Berkovich's ...
9
votes
1answer
354 views

Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?

Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum $$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$ Now ...
6
votes
0answers
254 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
14
votes
1answer
752 views

$p$-adic Bott periodicity?

The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...