Questions tagged [rigid-analytic-geometry]
rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields
18 questions
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Are flat morphisms of analytic spaces open?
Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map?
The ...
- 19.6k
45
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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
32
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1
answer
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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
27
votes
1
answer
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Motivation for relative schemes: why should one work with schemes over a ringed topos?
Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
- 11.8k
20
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3
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Étale homotopy type of non-archimedean analytic spaces
The following is likely all obvious to the experts. But since the field looks tricky to an outsider, maybe I may be excused for asking anyway.
I am wondering about basic facts of what would naturally ...
- 19.9k
18
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1
answer
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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
13
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1
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Reference for rigid analytic GAGA
I'm looking for a reference for the following result.
Theorem. Let $K$ be a complete, non-archimedean field, and let $X/K$ be a projective scheme, with analytification $X^\mathrm{an}$. Then the ...
- 1,838
12
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1
answer
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bornological vector spaces over a non-archimedean field
Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
- 1,180
10
votes
1
answer
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Analytic elements in non-archimedean geometry
Let $(k,|.|)$ be a complete non-archimedean valued field. Let $D$ be the open unit disc over $k$. (Anything I write could be adapted to the case of an open annulus.) The ring $\mathcal{O}(D)$ of ...
- 4,132
9
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1
answer
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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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8
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1
answer
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Are maps corresponding to affinoid subdomains flat in the Banach sense?
$\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\abs}[1]{\lvert #1\rvert}\newcommand{\comptensor}{\mathbin{\hat{\otimes}}}$
Let $k$ be a complete non-archimedian field and let $X = \Sp(B)$ be a $k$-affinoid ...
- 1,153
5
votes
1
answer
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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 ...
- 541
4
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1
answer
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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
4
votes
0
answers
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
4
votes
1
answer
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
4
votes
1
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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
3
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0
answers
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Is the Čech complex of a coherent sheaf on a compact separated rigid analytic space admissible?
$\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}$Let $X$ be a compact, separated rigid $k$-analytic space over some complete non-archimedean field $k$. Then $X$ has a finite affinoid covering ...
- 1,153
3
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0
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Sheaf of power-bounded elements in rigid analytic geometry
Let $k$ be a field with a non-archimedean complete valuation $|\ |$, $X$ a reduced rigid analytic space over $k$. The presheaf $\mathcal{O}^0$ which to an affinoid $U$ of $X$ attaches the ring $\...
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