Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
256 questions
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Universally adhesive pairs and adic spaces
In Fujiwara-Kato's book, formal schemes and rigid spaces are always required to be t.u. adhesive in order to satisfy many nice properties. If $V$ is a valuation ring of rank one and $\varpi$ is a ...
- 417
4
votes
0
answers
125
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Coherent sheaves and Kiehl Theorem B for adic spaces
In Huber's paper "A generalization of formal schemes and rigid analytic varieties", he set up the theory of adic spaces and their structure presheaves. He also gave some conditions under ...
- 321
2
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0
answers
195
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Explicit description of the crystal associated to the universal object on a Lubin-Tate space
Let $X:=\mathrm{LT}_h$ be the Lubin-Tate space of the (unique) formal group law of height $h$ over $k:=\overline{\mathbb{F}}_p$. The adic generic fiber $X_{\eta}=\mathrm{LT}_{h,\eta}$ is then a rigid-...
- 29
1
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0
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47
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Pointwise bounded subvariety in a rigid tube
I encountered the following question when studying the cohomology of a char p variety, which is really outside of my area of expertise.
Notation: $\mathbb{F}=\mathbb{F}_p$ bar, $W$ its Witt ring, $K=W[...
- 21
3
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0
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227
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Is it within reach of current mathematics to decide whether Lagarias and Bernstein's conjugacy map fixes no end in $\Bbb P^1(\Bbb Q_2)$?
In their 2009 paper, Lagarias and Bernstein discuss the conjugacy map which is the unique homeomorphism $T$ on the $2-$adic field $\Bbb Q_2$ which fixes $0\mapsto0$ and topologically conjugates $x\...
- 426
0
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0
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53
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Isometric map of affinoid p-adic algebras
Let $A = \mathbb{Q}_p\langle t_1, \dots, t_n \rangle = \mathbb{Q}_p\langle T_1, \dots, T_n \rangle/J$ be an $p$-adic affinoid algebra generated by $t_1, \dots, t_n$ with its norm being the quotient ...
- 1,607
1
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0
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100
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Adelic Tate-module of an Abeloid variety
Let $A$ is an abelian variety over a non-archimedian field $K$ of characteristic $0$. Then it is known that the adelic Tate-module $T_{\hat{\mathbb{Z}}}A$ of $A$ is a free $\hat{\mathbb{Z}}$-module of ...
- 73
2
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0
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176
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Finite and faithfully flat morphisms of Huber Rings
I am currently studying the notes of Kiran Kedlaya for Arizona Winter School 2017 on Perfectoid Spaces.
Link : https://swc-math.github.io/aws/2017/2017KedlayaNotes.pdf
I am stuck at Exercise $1.1.6$, ...
4
votes
1
answer
274
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algebraic fundamental group of Raynaud generic fiber
Let $k$ be a perfect field of characteristic $p$. Let $X$ be a quasi-projective smooth variety over the Witt ring $W=W(k)$ ($K=\mathrm{Frac}(W)$). Let $\mathcal X$ be the $p$-adic formal completion of ...
- 471
2
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1
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178
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Isolated points of affine varieties over algebraically closed topological fields
Let $K$ be a Hausdorff topological field, and let $V/K$ be an affine variety.
Recently I have been thinking about properties of the topology that $V(K)$ inherits from $K$. I realized that I have ...
4
votes
1
answer
411
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Universal property of the disk algebra in terms of power-bounded elements
Set $A(D)$ the disk algebra, i.e. the algebra of all analytic functions on the open unit disk $D$ in the complex plane which extend continuously to the closed unit disk $\overline{D}$. This is a ...
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1
vote
1
answer
277
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Is the Tate algebra over a complete DVR regular and/or a UFD?
If $k$ is a nonarchimedean field, the Tate algebra $T_n(k) = k\langle T_1,\dots,T_n\rangle$ is a regular UFD. If I replace $k$ with a complete DVR $R$, is the affinoid algebra $R\langle T_1,\dots,T_n\...
- 151
5
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0
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391
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Poincaré duality for smooth affinoid dagger spaces
I am trying to understand the proof of Poincaré duality for smooth affinoid dagger spaces in Große-Klönne's paper Rigid analytic spaces with overconvergent structure sheaf (https://arxiv.org/pdf/1408....
- 117
1
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0
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149
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Fiber dimension theorem for morphisms of topologically finite type
Is there analogue for fiber dimension theorem for topologically finitely type morphisms? I wish to know at least the following very special case:
Let $R$ be a sufficiently good commutative ring. Say $...
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3
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0
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225
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Colimits in commutative Banach algebras?
Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
- 526
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0
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141
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Krull dimension of affinoid algebra
Let $K$ be a complete field w.r.t. a valuation, with residue field $k$. Let $A$ be an affinoid algebra over $K$ with respect to a valuation $V$ (in the sense of Tate; in the terminology of Berkovich, $...
- 237
1
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1
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300
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Can a p-adic ball cover a p-adic ball?
Are there a polynomials $f_1,...f_n \in \mathbb{Z}_p[x_1,...x_n]$ with there coeficients $p$-adic integers s.t.
A map $F:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p^n$ defined by $f_1,...f_n$
satisfy the ...
- 378
3
votes
1
answer
332
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The simply connectedness of $\mathbb{A}^n_{\mathbb{Q}_p}$
My question is how to prove the affine $n$-space over $p$-adic number $\mathbb{Q}_p$ is simply connected.
To be precise,
Let $X$ be $p$-adically analytic manifold, $f:X\rightarrow \mathbb{A}^n_{\...
- 378
2
votes
1
answer
482
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Uniqueness and existence of maps
I am currently reading the Berkeley lectures on Perfectoid Spaces by Scholze and Weinstein. In the section "The adic open unit disk over $\mathbb{Z}_p$" we encounter from Proposition 4.2.6 ...
7
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0
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232
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Finite generation and finite presentation over a truncated valuation ring: is there an easier proof?
Let $K^+$ be a valuation ring which is $\pi$-adically complete for some pseudouniformizer $\pi$.
Nagata 053E proved that every finitely generated and flat (equivalently, torsion-free) $K^+$-algebra is ...
- 16.9k
1
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0
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216
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Galois action on the cohomology of a curve over a local field with bad reduction
Let $C/\mathbb Q_p$ (or a p-adic local field more generally) be a smooth projective curve with split semistable reduction over $\mathbb Z_p$. What can we say about the action of the Galois group $\...
- 8,071
4
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1
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605
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Help with understanding a rigid geometry proof
I am trying to read Coleman's paper "$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 ...
- 141
1
vote
0
answers
124
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Points on a rigid analytic variety and "points" on a formal model
Let $k$ be a finite extension of $\mathbb{Q}_p$. Let $X$ be a quasi-compact, quasi-separated rigid analytic variety over $k$. We choose a formal model $\mathcal{X}$ of $X$ over $\mathcal{O}_k$.
If I ...
- 539
3
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1
answer
241
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Reference Request: Preservation of étale maps under rigid analytic GAGA
Let $K$ be a finite extension of $\mathbb{Q}_p$. As the title says, I am looking for a reference in which it is shown that given an étale map $f:X\rightarrow Y$ between smooth algebraic $K$-varieties, ...
- 564
2
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0
answers
162
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Quasi-Stein spaces that are not Stein or affinoid
I'm looking for an example of a rigid analytic space, over a field containing the $p$-adic numbers which is complete with respect to a non-archimedean norm, that is quasi-Stein, in the sense of Kiehl, ...
- 117
4
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1
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366
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Chevalley's theorem on valuation spectra
In the paper On Valuation Spectra (Section 2, Page 176), Huber and Knebusch asserted that: if the ring map $A\to B$ is finitely presented then the associated map of valuation spectra $\mathrm{Spv}(B)\...
- 265
6
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0
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223
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Resolution property in rigid analytic geometry
I am not a rigid analytic geometrer, 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 ...
- 1,322
5
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0
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531
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) $ ...
- 151
11
votes
1
answer
475
views
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\...
- 1,000
9
votes
1
answer
2k
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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 $\...
- 103
6
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1
answer
551
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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 ...
- 1,235
4
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0
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235
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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....
- 4,242
4
votes
1
answer
418
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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
4
votes
1
answer
308
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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 $\tilde X$ (...
- 1,041
3
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0
answers
100
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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 ...
- 3,366
1
vote
1
answer
194
views
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 $\mathscr{O}_K$, ...
- 95
0
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0
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154
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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 $...
- 1,041
4
votes
0
answers
137
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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 ...
- 1,068
3
votes
1
answer
249
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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 ...
- 3,366
5
votes
0
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238
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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 ...
3
votes
1
answer
221
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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 ...
1
vote
0
answers
104
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) ...
- 36
1
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0
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98
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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}}{\...
- 1,041
5
votes
0
answers
150
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 ...
- 1,607
4
votes
1
answer
237
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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 ...
- 1,135
3
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0
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223
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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 ...
- 1,135
5
votes
0
answers
595
views
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
1
vote
1
answer
148
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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, ...
- 218
4
votes
0
answers
266
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. ...
- 583
3
votes
1
answer
629
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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 ...
- 31