# Questions tagged [rigid-analytic-geometry]

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

181
questions

2
votes

0
answers

72
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 $...

3
votes

0
answers

76
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

136
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

132
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

134
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

135
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 $k$-...

1
vote

0
answers

42
views

### Is the complement of an $\epsilon$-neighborhood an affinoid open polydisc in a flag variety?

$\newcommand{\rig}{\mathrm{rig}}$Let $K$ be non-Archimedean local field of characteristic zero with non-Archimedean norm $|\,\,|$ (assumed to be normalised), ring of integer $\mathcal{O}_K$ and $C$, ...

2
votes

0
answers

148
views

### Affinoid domain?

$\newcommand{\rig}{\mathrm{rig}}$Let $p$ be a prime and $K:=\mathbb{Q}_p$ with ring of integer $\mathcal{O}_K=\mathbb{Z}_p$. Further let $X$ be a projective variety. The associated rigid analytic ...

3
votes

0
answers

106
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

153
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

624
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

175
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 ...

2
votes

0
answers

283
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

69
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

275
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

253
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

343
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 ...

37
votes

2
answers

2k
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

342
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

332
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

372
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

917
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

232
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 ...

2
votes

1
answer

146
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 ...

10
votes

0
answers

209
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

144
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

208
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

219
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

871
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 ...

2
votes

0
answers

119
views

### Rigid \'etale cohohomology of flag variety minus its rational points e.g $p$-adic Drinfeld half plane

Let $Fl=G/B$ over $\mathbb Q_p$ be the flag variety of a quasi-split reductive group $G$ over $\mathbb Q_p$, then $X=Fl-Fl(Q_p)$ shall exist as a rigid analytic variety over $\mathbb Q_p$, how to ...

1
vote

0
answers

69
views

### Completion of $K$-algebra of finite type with respect to the residue norm

Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, ...

4
votes

0
answers

194
views

### nearby cycles map for affine formal schemes

Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\...

2
votes

1
answer

193
views

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

Let $k$ be a non-archimedean field and denote by $\mathbb{A}_k^n$ the analytic affine space of $n$ dimensions over $k$ (analytic in the sense of Berkovich). There is a natural injective map of sets $\...

7
votes

0
answers

207
views

### Etale maps from smooth affinoids to balls

Let $X$ be a smooth affinoid rigid space over a nonarchimedean field $K$. Does $X$ admit an etale map to a $K$-affinoid ball? The answer is almost certainly "no" in general, but I don't know ...

4
votes

1
answer

298
views

### Topological and algebraic covering spaces in Berkovich geometry

Let $k$ be a complete, non-archimedean field, and $X$ a Berkovich space over $k$ (as nice as you like, for arguments sake let's say strictly $k$-analytic, good, and geometrically connected). As ...

4
votes

1
answer

326
views

### Identity theorem in $p$-adic geometry/analysis

If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $\mathbb{Q}_p$ is disconnected. The modern approach to this is,...

6
votes

0
answers

254
views

### Rigid analytic varieties vs rigid spaces

In rigid analytic geometry, some sources refer to "rigid spaces", where others refer to "rigid analytic varieties". Do these two terms stand for the same thing, or is there a ...

12
votes

1
answer

417
views

### Can a covering space of the $p$-adic disc split over the circle?

Let $D = {\rm Sp}\, \mathbb{C}_p\langle x\rangle$ be the affinoid unit disc over $\mathbb{C}_p$.
Is there an example of a connected finite etale cover of $D$ whose restriction to the "unit circle" ${\...

6
votes

0
answers

337
views

### Coherent cohomology of the generic fiber of Lubin-Tate space vs. of Lubin-Tate space considered rationally?

I am trying to compare the coherent cohomology of the generic fiber of Lubin-Tate space to the coherent cohomology of Lubin-Tate space considered rationally, and I am going in circles! I would be very ...

2
votes

1
answer

239
views

### Reduced complete Tate ring which is not uniform?

Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...

4
votes

0
answers

95
views

### Existence of a "p-adic Mahler measure" or alternatively, the converge of a p-adic sequence

Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence:
$$a_n = \frac{1}{p^{n-1}}\...

1
vote

0
answers

131
views

### L-function in p-adic spaces

I've been learning more about different $p$-adic geometries, namely Berkovich spaces, Huber's Adic spaces and ridgid analytic spaces. In arithmetic geometry, it is often very interesting to assoicate ...

5
votes

1
answer

445
views

### Schemes over (locally) ringed spaces: working over complex-analytic spaces, rigid-analytic spaces, formal schemes, etc

Monique Hakim developed in her doctoral thesis [1] the theory of relative schemes. These comprise, as a special case, the theory of schemes over (locally) ringed spaces. What makes the study of these ...

24
votes

1
answer

969
views

### 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 ...

3
votes

0
answers

151
views

### gluing Berkovich spaces

In his paper Etale cohomology for non-Archimedean analytic space (IHES), Berkovich explained how to glue $k$-analytic spaces (Prop. 1.3.3) and show its uniqueness using the Prop 1.3.2 (gluing morphism)...

5
votes

0
answers

98
views

### Are affinoid algebras over nontrivially valued fields Jacobson?

It is well-known that for any field $k$ with valuation the Tate algebra $k\{T_1,\dots,T_n\}$ is Jacobson (see Bosch-Güntzer-Remmert for nontrivial valuations; for trivial valuations those are just ...

8
votes

1
answer

879
views

### Translation between formal geometry and rigid geometry

I'm reading a paper that translates between formal geometry and rigid geometry.
In particular, this paper begins with two rigid analytic spaces $A$ and $C$ (each coming from a scheme over $\mathbb{Z}...

6
votes

0
answers

216
views

### Is being perfectoid a local property?

In his 2012 CDM proceedings, Peter Scholze mentions the following open question:
Let $K$ be a perfectoid field and $(A,A^+)$ a complete affinoid $K$-algebra. Suppose there exists a cover of $X = \...

2
votes

0
answers

198
views

### Rigid analytic geometry and Tate curve

I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-...

8
votes

0
answers

175
views

### Looking for basic example in the theory of adic spaces

I have heard that there exist examples of Huber pairs $(A,A^{\circ})$ and $(B,B^+)$ such that $\operatorname{Spa}(B,B^+)$ is a rational open of $\operatorname{Spa}(A,A^{\circ})$ and such that $B^+ \...