Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
243 questions
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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 ...
- 893
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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| = ...
- 153
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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,...
- 4,418
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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
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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
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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
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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 ...
- 11.8k
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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
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438
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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) $ ...
- 151
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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 ...
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132
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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 ...
- 1,485
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556
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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 $...
- 51
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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 ...
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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 :...
- 37k
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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 ...
user140765
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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
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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 $\...
- 892
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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
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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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510
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Help with understanding a rigid geometry proof
I am trying to read Coleman's paper "$p$-adic Banach Spaces and Families of Modular Forms". In Proposition A5.5, he considers a quasi-finite morphism $f:Z\to Y=\mathbb{B}^1_K$ where $Z$ is a ...
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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 ...
- 101
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344
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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{...
- 913
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225
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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 ...
- 7,746
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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 ...
- 2,789
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Iwasawa logarithm and analytic continuation
I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let $\mathbb{C}_p$ be the completion of the algebraic closure of $\mathbb{Q}_p$.
...
- 539
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How should we understand the relative interior in Berkovich spaces
I'm reading Berkovich's book on analytic spaces. The notion of relative interior confuses me. Is there anyway to see how it "looks like"? For instance, if $r <1$, what is the relative interior of
\...
- 125
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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) ...
- 473
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218
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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$ (...
- 785
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545
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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 ...
- 541
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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}\}\...
- 541
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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 ...
- 1,838
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279
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Non-isomorphic varieties over a $p$-adic field with isomorphic analytifications
There are examples of non-isomorphic algebraic varieties defined over complex numbers whose associated complex-analytic spaces are isomorphic. Are there such examples for varieties defined over $p$-...
user138661
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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)...
- 910
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469
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Spherical completions and flatness
Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? ...
- 1,180
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1
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184
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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 ...
- 658
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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\...
- 541
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459
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Motivic cohomology of rigid analytic spaces
There is a satisfactory theory of B1-homotopy theory for rigid analytic spaces defined by Ayoub in the style of Voevodsky, and I'm aware of some work about the corresponding theory of motives, e.g. ...
- 2,054
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483
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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$...
- 2,789
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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
4
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1
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307
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scheme of generalizations
Hi,
I have the following problem. Let $\mathcal{O}$ be a valuation ring and $S=Spec(\mathcal{O})$, denote with $s$ the closed point and with $\eta$ the generic one. Let $X\rightarrow S$ be a proper, ...
- 43
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117
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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 ...
- 984
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205
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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. ...
- 563
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321
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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 ...
- 773
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149
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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 $...
- 119
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364
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(Co)limits of adic spaces
Which (co)limits exist in the category of adic spaces ? Also, can we impose adjectives such as "noetherian" or "quasi-compact", etc., to get more (co)limits ? I know that finite ...
- 1,516
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843
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An attempt to define partial properness and compactification for some maps between analytic spaces
The paper Étale cohomology of diamonds defines partial properness and compactification for maps between v-sheaves, and in particular for perfectoid spaces and rigid-analytic spaces. Recently when ...
4
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0
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281
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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],\...
- 1,093
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130
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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}}\...
- 7,746
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307
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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 ...
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 ...
- 41