# Questions tagged [rigid-analytic-geometry]

rigid analytic varieties, affinoid varieties, strictly convergent power series over non-archimedean fields

202
questions

8
votes

0
answers

380
views

### 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 ...

1
vote

0
answers

135
views

### Contractibility of the quotient of an analytification of a smooth variety by a finite group (if the field is trivially valued)

Let $k$ be a field and $X$ be a smooth irreducible $k$-variety with an action of a finite group $G$. I consider $k$ as a trivially valued field.
It is known from results of Berkovich ("Smooth p-...

1
vote

1
answer

80
views

### 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 ...

2
votes

0
answers

177
views

### The closed unit adic disk

I am reading the Scholze-Weinstein Berkeley lecture notes on "Perfectoid Spaces", and in particular I am stuck trying to understand the closed adic unit disk, which is the second example of ...

0
votes

1
answer

103
views

### 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 ...

3
votes

1
answer

328
views

### "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 ...

5
votes

1
answer

239
views

### 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. ...

4
votes

0
answers

250
views

### Can we see the completion of a scheme along a subscheme as an adic space?

Classically, formal schemes were invented to study completions of schemes along closed subschemes. Eventually, people started using them for more arithmetical reasons. (I.e., to study non-archimedean ...

0
votes

0
answers

56
views

### Weierstrass subdomain of $\DeclareMathOperator\Spm{Spm}\Spm \mathbb{Q}_p$

I am trying to understand Weierstrass subdomains of $\Spm\DeclareMathOperator\QP{\mathbb{Q}_p}\QP$.
Recall that a Weierstrass algebra of an affinoid space $\Spm A$, where $A$ is a Banach algebra with ...

1
vote

0
answers

128
views

### The stack $\operatorname{GL}_2/B$

Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...

4
votes

1
answer

176
views

### On inverse limits of $\pi$-adically complete algebras

Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...

4
votes

1
answer

408
views

### On the local properties of rigid analytic varieties

Let $K$ be a non-archimedean field complete with respect to a discrete valuation with ring of integers $\mathcal{R}$, uniformizer $\pi$ and residue field $k$. Consider an affinoid analytic $K$-variety ...

18
votes

1
answer

2k
views

### 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 ...

3
votes

1
answer

147
views

### On the stability of having a normal formal model under finite extensions of the base field

Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine ...

8
votes

1
answer

280
views

### 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}...

2
votes

0
answers

113
views

### Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$

Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...

5
votes

1
answer

325
views

### 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 ...

2
votes

1
answer

152
views

### Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)

Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering
$$ \dotsb \...

8
votes

1
answer

339
views

### Noetherian but not strongly Noetherian

What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but not strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \...

4
votes

0
answers

120
views

### Coherence of the I-adic completion of a local ring of a formal scheme

Let $K$ be a valued field of rank one and $K^+$ its valuation ring such that $K^+$ is $\varpi$-adically complete with respect to a pseudo-uniformizer $\varpi\in K^+$. Let $X$ be a smooth finite type $...

3
votes

1
answer

134
views

### Bounded torsion of quotients of affine formal models

$\DeclareMathOperator\Sp{Sp}$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 ...

4
votes

1
answer

324
views

### On a consequence of the Gerritzen-Grauert Theorem

Let $K$ be a local field of characteristic zero and $X$ an affinoid rigid space over $K$. Let $U\subset X$ be an affinoid subdomain, and consider a finite family of points $\{p_{1},\cdots, p_{n}\}\...

2
votes

1
answer

277
views

### Overconvergent modular forms and the level at $p$

I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.
The ...

2
votes

0
answers

134
views

### Theorem on formal functions when the initial data is a proper map of formal schemes

Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$.
Set $S_0=\{x\}$ be a closed point of $S$ and $...

5
votes

1
answer

136
views

### An example where the non-Archimedean tensor product of normed modules is only seminormed?

Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}_{\geq 0}$ which satisfies
$$ | 0_R| = ...

4
votes

1
answer

170
views

### Higher direct image of coherent sheaf and rigid analytification

Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$
be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) ...

3
votes

1
answer

200
views

### On the exactness of some completed tensor products

Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/...

2
votes

0
answers

173
views

### Enlightening examples of tropical skeletons of Berkovich spaces

Let $K$ be a complete non-archimedean field and let $X$ be a $K$-analytic space in the sense of Berkovich of pure dimension $d$. Let $\varphi \colon X \to \mathbf{G}_m^r$ be a moment map to an ...

1
vote

1
answer

160
views

### Reference request: Gruson's theorem on the tensor product of Banach spaces over a non-Archimedean field

I am looking for a reference for theorem 3.21 of these notes: https://web.math.princeton.edu/~takumim/Berkovich.pdf
The theorem states that if $k$ is a non-Archimedean field and $X$ and $Y$ are $k$-...

3
votes

0
answers

125
views

### Reference for a p-adic analytic Douady space

I am almost sure that some paper was published in German probably in the 60's or in the 70's proving the existence of a "p-adic analytic Hilbert scheme" (or Douady space) related to a given ...

2
votes

0
answers

199
views

### Differential forms on rigid analytic/adic spaces

What the correct notion of "Kähler differentials" on a sufficiently nice adic spaces (rigid space, perhaps) ? Given, a smooth variety $X$ over a perfect field $k$ of some positive ...

6
votes

1
answer

690
views

### Vector bundles on the various sites of a preperfectoid

Let $X$ be a preperfectoid space over $\mathrm{Spa}(\mathbb{Q}_p,\mathbb{Z}_p)$. It has several associated sites, with successively finer topologies: $$X_{an} \subset X_{et} \subset X_{proet} \subset ...

4
votes

0
answers

226
views

### (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 ...

4
votes

0
answers

604
views

### An attempt to define partial properness and compactification for some maps between analytic spaces

The paper Étale cohomology of diamonds defines partial properness and compactification for maps between v-sheaves, and in particular for perfectoid spaces and rigid-analytic spaces. Recently when ...

2
votes

0
answers

105
views

### Computing the ring of power-bounded elements in an affinoid algebra

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $A$ be an affinoid $K$-algebra, i.e. $A$ is isomorphic to a quotient of the Tate algebra $K\left<T_1,\dotsc,T_n\right>$ for some $n$. ...

4
votes

1
answer

338
views

### Motivic cohomology of rigid analytic spaces

There is a satisfactory theory of B1-homotopy theory for rigid analytic spaces defined by Ayoub in the style of Voevodsky, and I'm aware of some work about the corresponding theory of motives, e.g. ...

3
votes

1
answer

287
views

### Geometric line bundles on the Tate curve

Let $E_q$ be the rigid analytic Tate elliptic curve over a complete algebraically closed non-archimedean field $K$ of mixed characteristic $(0,p)$, with parameter $q\in K^{\times}$ with $|q|<1$.
...

3
votes

1
answer

373
views

### Why does $\mathbb C_p$ not contain the periods?

I am reading the following article of Berger, p8 and I don't understand the idea:
$C_p:=\widehat{\overline{\mathbb Q_p}}$ does not contain the periods
The text seem to reason as follows
(under some ...

43
votes

2
answers

3k
views

### 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-...

9
votes

0
answers

366
views

### Why do Coleman functions form a sheaf?

In section 4 of Ammon Besser's 2002 'Coleman Integration Using the Tannakian Formalism,' he defines abstract Coleman functions, which we can describe roughly as those functions which arise by iterated ...

0
votes

0
answers

385
views

### Analytic structures on the source of a surjection of condensed rings

Question. Let $(\mathcal B,\mathcal N)$ be an analytic (animated associative) ring, $\mathcal A$ be a condensed (animated associative) ring and $f\colon\mathcal A\to\mathcal B$ a surjective map of ...

12
votes

1
answer

439
views

### Open immersion of affinoid adic spaces

If $R$ and $S$ are complete Huber rings with $\varphi: R \to S$ a continuous map, then is it true in general that if $\mathrm{Spa}(S, S^\circ) \to \mathrm{Spa}(R, R^\circ)$ is an open immersion of ...

6
votes

2
answers

1k
views

### Vector bundles on adic spaces

Let $X = \mathrm{Spa}(A,A^+)$ be an analytic sheafy adic space. Let $\mathcal{E}$ be a locally finite free $\mathcal{O}_X$ sheaf. Does $\mathcal{E}$ correspond to a geometric vector bundle over $X$? ...

6
votes

0
answers

275
views

### $F$-isocrystals defined via a lift of a scheme

Let $X$ be a smooth affine scheme over a finite field $k$. Then there exists a smooth affine formal scheme $\mathfrak{X}$ over $W(k)$ with a lift $\sigma$ of the Frobenius. A convergent $F$-isocrystal ...

3
votes

1
answer

207
views

### complement of "good reduction" points in p-adic shimura varieties

assume that $X$ is Siegel Shimura variety defined over $\mathbb{Z}_p$, you can take its p-adic formal completion $\mathfrak{X}$,and than take it's adic generic fiber $\mathcal{X}$ and get an adic ...

11
votes

0
answers

294
views

### Quasi-separated rigid-analytic space without a formal model?

Well, my question is slightly embarrassing. When learning rigid geometry (mostly from Bosch's book) I realized that I don't know the answer to the following basic question.
Question. Is there an ...

1
vote

0
answers

161
views

### Moduli interpretation of normalization of moduli space

The question is about formal and rigid geometry, but I would be interested in an answer from an algebraic geometry point of view as well.
Let $\mathfrak{X}$ be a formal moduli space (e.g., the formal ...

3
votes

1
answer

253
views

### How to show analytification functor commutes with forgetful functor?

Let $k$ be a field complete with respect to a non-trivial non-archimedean
absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$.
Denote $X\rightsquigarrow X^{...

7
votes

1
answer

268
views

### Indeterminacy locus of meromorphic maps of rigid analytic spaces

Setup. Let $k$ be an algebraically closed field of characteristic zero. Let $X/k$ be a normal variety, and let $Y/k$ be a proper variety. It is well-known that the indeterminacy locus of a rational ...

9
votes

1
answer

1k
views

### On the definition of the etale site of an adic space

I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".
First ...