Questions tagged [berkovich-geometry]
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71 questions
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Is there a geometric realization of $\mathbf{C}((t))$-varieties?
Let $MV_F$ be the $\mathbf{A}^1$-homotopy category over the field $F$. Let $H$ be the homotopy category of spaces, and let $H_{/S^1}$ be the homotopy category of spaces over the circle.
When $F = \...
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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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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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2
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Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids
Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ ...
- 892
6
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1
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Relations between two definitions of non-archimedean analytic spaces
I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature.
Let us fix a non-archimedean complete valuation field $...
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4
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2
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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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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
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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 $\...
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5
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1
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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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applications of Berkovich spaces
What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$
user19475
19
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2
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Open problems in Berkovich geometry
I would like to know if there is a state of the art recent reference on non-archimedean analytic spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (*). I have ...
- 255
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1
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397
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The tensor product of admissible morphisms of semi-normed modules over a normed ring is an admissible morphism (V. G. Berkovich)
Disclaimer : I found here https://mathoverflow.net/editing-help in the spoilers paragraph that putting >! would hide following things, which was a way for me to alleviate my question's presentation by ...
- 255
3
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1
answer
560
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is every point of a Berkovich space a Shilov point?
Let $k$ be an algebraically closed non-Archimedean valued field with the value group $\mathbb R$, and let $X$ be a variety over $k$. Is it true that for any point $x \in X^{an}$ of the Berkovich ...
- 4,070
2
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1
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The target of a regular function in Non-archimedean analytic geometry
Let $(k,|\cdot|)$ be an algebraically closed field, complete wrt a (multiplicative) norm as in the framework of the Berkovich's analytic geometry. Given a commutative Banach $k$-algebra $\mathcal{A}\...
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0
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A functor of points approach to Berkovich analytic spaces
Is it possible to define a Berkovich analytic space via its functor of points?
Let $k$ be a complete non-Archimedean field, possibly the trivial one. I am tempted to define a Berkovich analytic space ...
- 171
7
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Etale cohomology of Berkovich spaces
Suppose $X/\mathbb{Q}$ is a reasonable smooth projective variety with interesting etale cohomology. For example, we can say $X$ is an elliptic curve. To what extent does it make sense to study the ...
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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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Rigid analytic spaces vs Berkovich spaces vs Formal schemes
I wonder if someone could explain briefly what is the relation between these 3 formal models, of a Berkovich space, a rigid analytic space and a formal scheme?
I have been working with formal schemes ...
- 251
14
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0
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771
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Cohesive ∞-toposes for analytic geometry
There is a class of big ∞-toposes that come with a good supply of intrinsic notions of differential geometry and differential cohomology: called cohesive ∞-toposes (after Lawvere's cohesive toposes).
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22
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3
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Higher dimensional berkovich spaces
I am looking for a geometric and topological way to make a visualization of higher dimensional berkovich spaces, starting with the berkovich plane. Of course, this is just a collection of bounded semi-...
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1
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Do Berkovich homogenous spaces exist?
Let G be a k-analytic group, and let H be a closed subgroup of G. Then does there exist a k-analytic space, which can be reasonably called the quotient G/H?
Commentary: I realise that I am not being ...
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