The berkovich-geometry tag has no usage guidance.

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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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**1**answer

249 views

### 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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**1**answer

180 views

### 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 $X = \operatorname{spec} \mathcal{A}$ ...

**5**

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**1**answer

158 views

### 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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**2**answers

159 views

### 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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**2**answers

187 views

### 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 ...

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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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173 views

### 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} \...

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573 views

### applications of Berkovich spaces

What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$

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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 ...

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**1**answer

169 views

### 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 http://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 ...

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**1**answer

222 views

### 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 ...

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**1**answer

114 views

### 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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### 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 ...

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368 views

### 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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417 views

### 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
\...

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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 ...

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567 views

### 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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### Higher dimensional berkovich spaces

I am looking for a geometric and topologic way to make a visualization of higher dimensional berkovich spaces, statring with the berkovich plane. Of course, this is just a collection of bounded semi-...

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**1**answer

1k views

### 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 ...