Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
243 questions
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1
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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, ...
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0
answers
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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 ...
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12
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1
answer
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(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 ...
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23
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3
answers
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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 ...
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0
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1k
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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{...
user92332
5
votes
0
answers
188
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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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0
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279
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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 $(U, \...
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1
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$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 ...
user113453
8
votes
2
answers
1k
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$p$-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 $\text{exp} : \text{Lie}(G)\to G$?
user113393
11
votes
3
answers
2k
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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 $...
user113393
3
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0
answers
275
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 ...
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3
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0
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331
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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 $...
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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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1
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0
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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 ...
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5
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2
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596
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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 ...
- 4,070
9
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0
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268
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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
...
- 4,070
2
votes
1
answer
329
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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
0
answers
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 $\mathbb{F}_q$, $q=p^m$, $...
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2
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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
6
votes
2
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323
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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 ...
- 4,070
4
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0
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422
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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 ...
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3
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0
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209
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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 ...
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5
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1
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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 ...
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votes
3
answers
7k
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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 (...
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1
answer
380
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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 ...
- 609
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 ...
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4
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1
answer
421
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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 ...
user96873
41
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2
answers
3k
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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 ...
- 843
5
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1
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584
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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 ...
- 1,066
5
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0
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1k
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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). ...
- 825
8
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1
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471
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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 ...
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1
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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 ...
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6
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0
answers
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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 ...
- 323
18
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1
answer
1k
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$p$-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 ...
- 3,017
4
votes
2
answers
622
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Is there a notion of pure dimension for Berkovich analytic space?
For affinoid spaces the definition is similar to algebraic geometry, what about general analytic spaces? I can't find a reference about it. If yes then is the analytification of a variety of pure ...
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0
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518
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$p$-adic uniformisation of abelian varieties
In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement:
Let $A$ over $\mathbf{Q}_p$ be an abelian variety ...
- 121
5
votes
1
answer
471
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Rigid analytic geometry in characterstic 0 vs positive characteristic
This question is motivated purely by curiosity. In algebraic geometry there is a major distinction between the world of characteristic $0$ and that of characteristic $p > 0$ with different methods, ...
- 912
6
votes
1
answer
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Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)
In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.
Could you ...
- 845
14
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0
answers
555
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Vanishing of rigid cohomology for affine varieties
Let $k$ be a perfect field of positive characteristic and denote by $K$ the field of fractions of the ring of Witt vectors over $k$.
Question: If $X$ is an affine variety over $k$, do the rigid ...
- 141
1
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2
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766
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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
10
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0
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409
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Detecting $k$-affinoid spaces by vanishing cohomology
The property of being an affine scheme can be tested against all quasi-coherent sheaves in the following sense: a noetherian scheme $X$ is affine iff $H^i(X,\mathcal{F}) = 0$ for all quasi-coherent $\...
- 582
3
votes
2
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524
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Product of reduced affinoid spaces over a field is reduced (reference request)
Let $K$ be a field of characteristic zero complete with respect to a non-Archimedean absolute value. Suppose that $A$ and $B$ are two affinoid $K$-algebras. I'd like a reference that will answer the ...
- 408
8
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1
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why don't (can't?) we sheafify the structure presheaf of an adic space
In the definition of an adic space, usually there is a presheaf defined by first saying what it is on a particular basis of the topology of the underlying space, the so called rational subsets. One ...
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11
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3
answers
1k
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Trivialisation of vector bundles on Stein spaces
Does every vector bundle on a Stein space have a finite local trivialisation?
Definitions:
Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...
- 1,450
4
votes
1
answer
685
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Paper of Boutot-Carayol in `Courbes modulaires et courbes de Shimura'
I am trying to obtain a copy of the following
J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les
théorèmes de Čerednik et de Drinfel'd , Astérisque No. 196-197 (1991)...
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5
votes
1
answer
333
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Topology of ring of global sections of finite union of affinoid opens in a rigid analytic space
Let $X$ be a rigid analytic space over a non-Archimedean field $k$. If $U_1,\ldots,U_n\subseteq X$ are affinoid opens, then it's usually not clear whether or not the admissible open $U=U_1\cup\cdots\...
- 3,380
5
votes
1
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367
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Berkovich stalk versus rigid analytic stalk
Let $A$ be a strictly affinoid algebra. Let $X^{Ber}$ bet its Berkovich spectrum and $X^{Tate} = \operatorname{Sp} A$ its affinoid variety in the sense of rigid analytic geometry. Let $\mathfrak{m} \...
- 1,450
1
vote
0
answers
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
5
votes
1
answer
1k
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The rigid-analytic open disk
Let $K$ be a local field and $D_K$ the open unit disk, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D_K$? Proposition 1.1 of this article describes ...
- 3,605
23
votes
1
answer
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function field analogy and global/absolute geometry
The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
- 19.9k