Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
243 questions
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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 $\...
- 426
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73
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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, $...
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1
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242
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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 ...
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3
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1
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254
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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_{\...
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2
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0
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153
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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 ...
4
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1
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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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203
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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 ...
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3
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1
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187
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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, ...
- 541
2
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0
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110
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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 $\...
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11
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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
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98
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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 ...
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3
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1
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316
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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)\...
- 255
2
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87
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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
7
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1
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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 $\...
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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) $ ...
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2
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0
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679
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Roadmap for p-adic geometry
I think some questions asked in similar fashion with this one. I am a master student in mathematics. I have knowledge in algebraic geometry(both in Shafarevich's and Vakil's books), algebraic topology ...
- 21
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155
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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 ...
- 1,129
1
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1
answer
131
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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 $\mathscr{O}_K$, ...
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302
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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
5
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1
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318
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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 ...
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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 $\tilde X$ (...
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3
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1
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180
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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 ...
3
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0
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146
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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....
- 4,164
2
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1
answer
226
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Restrictions of affinoid functions from wide open neighbourhoods
Let $X=\operatorname{Sp}(A)$ be an affinoid $K$-space, where $K$ is a p-adic field. Suppose that $X$ lies in the interior of another affinoid $K$-space $X'=\operatorname{Sp}(B)$. Recall that this ...
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1
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200
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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 ...
- 2,907
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197
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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
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0
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82
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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 ...
- 2,907
4
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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 ...
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4
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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 ...
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1
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Why are there three kinds of non-archimedean geometry?
It may seem silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to ...
- 1,270
5
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0
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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 $...
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111
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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 $...
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0
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183
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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 ...
- 658
5
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0
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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
3
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1
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469
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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
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1
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370
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G-topological spaces and locales
Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
user513306
0
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1
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287
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When is the power-bounded subring top. of finite type?
Very naive question here. Let $K$ be a complete nonarchimedean field, $A$ a reduced affinoid $K$-algebra. When is the power-bounded subring $A^\circ$ topologically of finite type, in the sense that we ...
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2
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Are rigid-analytic spaces obsolete, since adic spaces exist?
Recently in a seminar the following question was raised and, despite my familiarity with theory, I couldn't come up with a good answer:
Are there any good reasons to use Tate's theory of rigid-...
- 28.2k
1
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1
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98
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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, ...
- 208
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0
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80
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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}}{\...
- 785
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0
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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. ...
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9
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0
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578
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What lies between algebraic geometry and analytic geometry?
Algebraic geometry and analytic geometry are closely related (witness GAGA). But the latter still seems much "bigger" than the former. I'd like to be able to get from algebraic geometry to ...
- 64k
3
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1
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504
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"Non-algebraic" Berkovich spaces
Usually, Berkovich analytic spaces are derived from some Banach rings (or chains of Banach rings) over a completely normed field $k$ through Berkovich spectrum. But when the base field is the complex ...
- 1,543
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339
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On actions of finite groups on adic spaces
Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
- 541
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1
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133
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Injectivity of sheaf restriction maps for wide open neighbourhoods of rational subdomains
Let $X=\text{Sp}(A)$ be an affinoid $K$ space, where $K$ is a $p$-adic field. If $f_0, f_1,..., f_s \in A$ generate the unit ideal then we can define the rational subdomain $U= X(f_0, f_1..., f_s) = \{...
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297
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What's the relation between pseudo-compact and admissible rings?
We recall two definitions. Let $A$ be a linearly topologized ring which is complete and Hausdorff.
We say that $A$ is pseudo-compact if, for every open ideal $I\subset A$, the ring $A/I$ is artinian. ...
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1
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111
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Complete residue field of a point of type 5
Let $(F,|.|)$ be a complete algebraically closed field. Let $x$ be the point of type 5 corresponding to the unit open disc of the adic affine line over $F$. Can we obtain a concrete description of the ...
1
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0
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84
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Algebraizable image of a morphism of Galois cohomology stacks
Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...
- 2,907
1
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0
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86
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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
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362
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On the noetherianess of some subalgebras of an affinoid algebra
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
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