28
votes

### Are rigid-analytic spaces obsolete, since adic spaces exist?

My opinion (knowing that this is somehow a matter of taste) is that the rigid-analytic viewpoint is more or less obsolete, and that one should better use Berkovich, Huber or Raynaud instead (each of ...

27
votes

Accepted

### Are rigid-analytic spaces obsolete, since adic spaces exist?

There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications?
For the first question, it'...

20
votes

Accepted

### Berkovich space including both archimedean and non-archimedean worlds

The definition of analytic space over $\mathbf{Z}$ was given by Berkovich in his foundational book "Spectral theory and analytic geometry over non-Archimedean fields" (see the beginning of section 1.4 ...

18
votes

Accepted

### applications of Berkovich spaces

I would first recommend the paper of Antoine Ducros (Espaces analytiques $p$-adiques au sens de Berkovich, Séminaire Bourbaki, exposé 958, 2006) for a general survey of the theory, with applications.
...

16
votes

### Is there a geometric realization of $\mathbf{C}((t))$-varieties?

I know nothing about $\mathbf{A}^1$-homotopy. I will describe how one can get a functor from smooth varieties over $\mathbf{C}((t))$ to ${\rm Top}_{/\mathbf{S}^1}$ using log smooth proper models and ...

13
votes

Accepted

### Open problems in Berkovich geometry

This is a very broad question and it is difficult to know where to start. Remember that Berkovich's theory is a theory of analytic geometry, hence it makes sense to look for the counterpart of ...

13
votes

### Open problems in Berkovich geometry

I do not know if this falls within the scope of your question, and moreover I do not have a specific reference to point to, but there are certainly plenty of unsolved questions involving the dynamics ...

10
votes

### Is there a notion of pure dimension for Berkovich analytic space?

Yes, there is a good notion of dimension, due to Berkovich and developed in my article, as mentioned in the two answers above.
Concerning your question about GAGA principle for pure dimension, the ...

9
votes

### Berthelot functor, rigid analytic space

The setup of the question is not general enough:
(i) you mean to work with Spf rather than Spec,
(ii) Raynaud's construction doesn't apply to the formal scheme Spf($A$) for such $A$,
(iii) the ...

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8
votes

### Is there a geometric realization of $\mathbf{C}((t))$-varieties?

I don't know about $\mathbb A^1$-homotopy categories. Let me try to do something that I think will work just for the category of quasiprojective varieties.
The key lemma is this:
There is a ...

7
votes

### applications of Berkovich spaces

Let me add a few more applications to what has already been mentioned.
Relation with Bruhat-Tits buildings (Berkovich, then Rémy/Thuillier/Werner). If $G$ is a reductive group over a non-archimedean ...

6
votes

Accepted

### Abel-Jacobi map for Mumford curves analytically

This is done in Manin and Drinfeld's "Periods of p-adic Schottky groups." Journal für die reine und angewandte Mathematik 0262_0263 (1973): 239-247. You will also find it in Gerritzen and van der Put'...

6
votes

Accepted

### Are maps corresponding to affinoid subdomains flat in the Banach sense?

This is not true. Assume that $X$ is the closed unit disc (given by $|T| \le 1$, with algebra $B$) and $V$ is a smaller disc (given by $|T| \le r$ for some $r \in (0,1)$, with algebra $B_V$). Consider ...

5
votes

Accepted

### Relations between two definitions of non-archimedean analytic spaces

Let me give more a few more details than in nfdc23's comment. The most general definition of a analytic space is the one that you find in Berkovich's IHES paper. One requirement is that for every ...

5
votes

Accepted

### Good analytic spaces over a field into locally ringed spaces is fully faithful

First, I would like to say that I do not understand why you need the full faithfullness of the analytification functor. It seems to me that the main point is to prove that giving a morphism from an ...

5
votes

Accepted

### Is $\mathbb{A}_k^n(k)$ dense in the Berkovich analytification of $\mathbb{A}_k^n$?

If $k$ is not algebraically closed, then $\mathbb A^n_k(k)$ is not doing to be dense with $\mathbb A_k^n$. I'll show this for $n=1$ for simplicity. Take any point $P$ in $\mathbb A^1_k$ with a residue ...

4
votes

Accepted

### Why is the Berkovich spectrum of a C*-Algebra the same as the Gelfand spectrum?

Observe that every commutative C*-algebra $A \not\simeq \mathbb{C}$ has a non-trivial zero-divisor.
To see this view $A$ as continuous functions over its spectrum which contains at least two points (...

4
votes

### applications of Berkovich spaces

Jan Kiwi (Duke Journal) used Berkovich spaces to give the first proof of my conjecture that any sequence of quadratic rational maps which divergence in moduli space (of Mobius conjugacy classes) has ...

4
votes

### Berkovich space including both archimedean and non-archimedean worlds

You should take a look at the paper by Jérôme Poineau:
La droite de Berkovich sur Z, Astérisque n° 334 (2010)
and other papers by the same autor, like
Espaces de Berkovich sur Z : étude locale, ...

3
votes

### Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids

Yes, there is such a comparison theorem. It works not only for finitely generated $K$-algebras, but more generally for finitely generated $A$-schemes for $A$ an affinoid algebra. This is done by ...

3
votes

Accepted

### finite number of vertices of the polyhedron of variation of an invertible function on a Berkovich curve

It is not true in general that $P$ is finite. To see this, take an open disk $D$ and a non-zero function $f$ on it with infinitely many zeroes. In this case, your polyhedron of variation is infinite: ...

3
votes

### Abel-Jacobi map for Mumford curves analytically

You can also find an interpretation using integrals, this time multiplicative, and mesures, in the preprint "The Abel Jacobi map for Mumford Curves via Integration":
https://arxiv.org/abs/1609.09285
...

3
votes

Accepted

### Definition of model functions and their density in $C^0(X^\text{an})$

As alluded to in the question, the space $\mathcal{D}(X)_{\mathbb{Z}}$ is indeed the space of model functions arising from integral divisors.
Now, in order to deduce that $\mathcal{D}(X)_{\mathbb{Q}}$ ...

3
votes

Accepted

### is every point of a Berkovich space a Shilov point?

No. Saying that the point $x$ lies in the Shilov boundary of $\mathcal{M}(A)$ gives strong restrictions on the completed residue field $\mathscr{H}(x)$. In your case, its residue field $\widetilde{\...

3
votes

Accepted

### The target of a regular function in Non-archimedean analytic geometry

If you want to think of $f$ in a way not too far from complex intuition, you should rather consider the induced morphism $\varphi$ from $X$ to the Berkovich affine line. If $x\in X$ then $\varphi(x)$ ...

3
votes

Accepted

### An example of a morphism of rigid analytic spaces with affinoid base which is proper but does not satisfy $(\dagger)$

These two notions are actually equivalent (at least if $k$ is the fraction field of a dvr $R$), but I do not know any direct way to see this.
The proof I know heavily uses the theory of formal ...

3
votes

Accepted

### Completed tensor product is exact

To make things precise, let me add the end of the quoted sentence: "with admissible linear operators as morphisms". Moreover, I believe that Berkovich refers here to tensor products over a fixed base ...

3
votes

Accepted

### What is the definable functor associated to an algebraic scheme (model theory of valued fields)

This follows by elimination of imaginaries in algebraically closed fields. Given a finite affine cover of your scheme, one obtains a functor to Set by gluing the affine charts. This is indeed not ...

2
votes

Accepted

### Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids

Not sure if this is still relevant, but the desired comparison in the case of "principal interest" to you can be proved in two different ways.
On one hand, you can deduce it from some recent results ...

2
votes

Accepted

### Berthelot functor, rigid analytic space

Let me say something related to your first question (the local one) in a slightly different setting. Let $\mathscr{A}$ be a $\mathbb{Q}_p$-affinoid algebra and consider its reduction $\tilde{\mathscr{...

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