# Questions tagged [rigid-analytic-geometry]

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

147
questions

**12**

votes

**1**answer

277 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

284 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

133 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**

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**0**answers

63 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**

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**0**answers

98 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 ...

**4**

votes

**1**answer

374 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 ...

**21**

votes

**1**answer

748 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**

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**0**answers

122 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**

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**0**answers

77 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 ...

**7**

votes

**1**answer

614 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}...

**3**

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148 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

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142 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-...

**1**

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**0**answers

95 views

### Are restriction maps of Stein space isometries?

Let $(X,\mathcal{O}_X)$ be a rigid K-space, which is a Stein space in the sense of Kiehl i.e. $X$ has an admissible open covering by affinoids $\{U_i\}_{i \in \mathbb{N}}$ s.t. for all $i$
$U_i\...

**7**

votes

**0**answers

131 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^+ \...

**2**

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**0**answers

131 views

### What is meant by a “closed point” of a rigid analytic space?

This is perhaps a naive question, but I have seen some authors (such as Mazur, Kisin) use the term of a "closed point" for (classical) rigid analytic spaces. I am slightly confused as to what this ...

**2**

votes

**1**answer

171 views

### Čech complex of rigid $K$-space - Closed image of boundary maps

Let $(X,\mathcal{O}_X)$ be a rigid $K$-space with a finite affinoid covering $(U_i)_{i\in I}$ such that any intersection of the $U_i$ is affinoid too.
Equipping the direct products with the maximum ...

**1**

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**0**answers

49 views

### Complement of admissible open subset

Let $X$ be a rigid $K$-space and $U \subset X$ be admissible open. Is the complement of $U$ in $X$ again a rigid $K$-space?

**1**

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72 views

### Morphism into K-analytic projective space $\mathbb{P}_K^{n,rig}$

Let $K$ be a local field, with a completed algebraic closure $\hat{\bar{K}}$ and $A$ be an affinoid $K$-algebra. Is it then true, that for $f_0, \ldots ,f_n \in A$, the map $\phi:$ Sp$A \rightarrow$ $\...

**7**

votes

**1**answer

190 views

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

$\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\abs}[1]{\lvert #1\rvert}\newcommand{\comptensor}{\mathbin{\hat{\otimes}}}$
Let $k$ be a complete non-archimedian field and let $X = \Sp(B)$ be a $k$-affinoid ...

**6**

votes

**1**answer

314 views

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

Let $k$ be a complete non-archimedean field and let $\varphi \colon X \to Y$ be a morphism of rigid analytic spaces over $k$, where $\newcommand{\Sp}{\operatorname{Sp}}Y = \Sp(B)$ is affinoid. ...

**1**

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**0**answers

46 views

### Polytopal domains in non-archimedean torus

Given a non-archimedean field $\mathbb K$, there is a natural map
$$
\mathrm{val}: (\mathbb K^*)^n\to\mathbb R^n$$
(See Section 4 of Gubler's paper).
Gubler mentions there $\mathrm{val}$ is a ...

**5**

votes

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148 views

### Is this subset of a rigid space an admissible open?

Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball.
Consider the sets $U := \{ |T_1| < 1\}$ and $V :...

**3**

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173 views

### Is the Čech complex of a coherent sheaf on a compact separated rigid analytic space admissible?

$\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}$Let $X$ be a compact, separated rigid $k$-analytic space over some complete non-archimedean field $k$. Then $X$ has a finite affinoid covering ...

**4**

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127 views

### Rigid cohomology with support and dagger spaces

Let $K$ be a $p$-adic field with residue field $k$, and $X$ a variety over $k$. The rigid cohomology of $X$ over $k$ can be described very neatly using Grosse--Kloenne's notion of dagger spaces: embed ...

**1**

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90 views

### Quasi-coherent sheaves in Tate's theorem

A theorem of Tate states that for any affinoid space $X=\mathrm{Sp}\:A$ and a coherent sheaf $F$ we have $H^i(X,F)=0$ for all $i>0$.
Is this theorem true for merely quasi-coherent sheaves? The ...

**3**

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93 views

### A proper analytic surface into which every smooth proper analytic curve embeds

Let $k$ be a finite extension of $\mathbb{Q}_p$. Does there exist a proper $k$-analytic surface such that there is a closed immersion into it from any connected smooth proper $k$-analytic curve? The ...

**4**

votes

**1**answer

410 views

### Do coherent sheaves on rigid analytic spaces form an abelian category?

It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (...

**6**

votes

**2**answers

286 views

### What is the difference between total integral closure and integral closure?

I was advised here to make this a new question:
What is the difference between total integral closure and integral closure (geometrically, in the context of rigid analytic geometry)? I have read in ...

**4**

votes

**1**answer

147 views

### Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.
See for ...

**5**

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106 views

### Homotopy of rigid analytic spaces

Let $K$ be a complete non-Archimedean valued field (I think the valuation does not have to be discrete). For a paracompact strictly $K$-analytic space, I have seen at least two definitions of ...

**1**

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56 views

### Failure of Tate acyclicity for integral structure sheaves

Let $(A,A^+)$ be a sheafy Tate-Huber pair, and let $X=\operatorname{Spa}(A,A^+)$. It is well-known that $H^i(X,\mathcal{O}_X)=0$ for $i>0$. I assume it is generally not true that $H^i(X,\mathcal{...

**3**

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**1**answer

283 views

### Crystalline comparison for rigid-analytic varieties

Let $k$ be a finite extension of $\mathbb{Q}_p$. In this paper, Scholze proves an analogue of de Rham comparison for proper smooth rigid-analytic varieties over $k$. He also says:
...it should be ...

**8**

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**0**answers

332 views

### Foundational Questions on Adic Spaces

There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...

**2**

votes

**1**answer

205 views

### $p$-adic Kato--Nakayama space

Given a log scheme over $\mathbb{C}$ whose underlying scheme is locally of finite type, you can associate to it a ringed space called the Kato--Nakayama space. Is there a $p$-adic analogue of this ...

**8**

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275 views

### Serre's examples in rigid analytic geometry

Over $\mathbb{C}$, we have the following phenomenon: there exist algebraic varieties whose etale homotopy types are isomorphic but the homotopy types of their analytifications are not. Such examples ...

**4**

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**1**answer

199 views

### Non-isomorphic varieties over a $p$-adic field with isomorphic analytifications

There are examples of non-isomorphic algebraic varieties defined over complex numbers whose associated complex-analytic spaces are isomorphic. Are there such examples for varieties defined over $p$-...

**2**

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**1**answer

154 views

### Quasi-coherent sheaves on affinoid space

From Conrad's notes on rigid geometry:
More specifically, Gabber has given an example of a sheaf of modules $F$ on the closed unit disk $B^1$ such that $F$ is locally a direct limit of coherent ...

**6**

votes

**1**answer

284 views

### Rigid versus log-rigid cohomology for semistable varieties

If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ...

**8**

votes

**1**answer

459 views

### Voisin examples in $p$-adic geometry

Let $K$ be an algebraic closure of p-adic rationals. Does there exist a proper smooth rigid-analytic variety over $K$ whose etale homotopy type is not isomorphic to etale homotopy type of a proper ...

**6**

votes

**1**answer

417 views

### Flatness of the integral closure

Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$.
Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...

**1**

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**0**answers

121 views

### Status of locality of perfectoidness for uniform rings

Let $k$ be a perfectoid field of zero characteristic. Recall that a Tate $k$-algebra is called uniform if the set of power-bounded elements is bounded. Let $(A, A^+)$ be a uniform complete affinoid $k$...

**1**

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101 views

### Affinoid algebra and fundamental theorem of algebra

This post is closely related to the previous one here.
But more generally, we want to study an affinoid algebra $A:=T_n/\mathfrak a$. Let's assume $\mathfrak a= (f_1,\dots,f_r)$ for some $f_i\in T_n$....

**1**

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**0**answers

118 views

### Explicit description of rigid analytification of torus

It is known that in non-archimedean world there is also a GAGA-functor from the category of $K$-schemes of locally finite type to the category of rigid $K$-spaces. Here $K$ is a field with a non-...

**3**

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**1**answer

315 views

### Tate algebras and fundamental theorem of algebra

Let $\mathbb K$ be an algebraically-closed complete non-archimedean field whose absolute value is non-trivial. Consider the Tate algebra $T_n=\mathbb K\langle X_1,\dots, X_n \rangle$ and fix $f\in T_n$...

**3**

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190 views

### Etale cohomology of projective spaces in the rigid analytic setting

Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...

**5**

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220 views

### Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$

$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...

**32**

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**0**answers

821 views

### Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...

**10**

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233 views

### Zeros of $p$-adic power series and rationality

Let $K$ be a non-archimedean field with valuation ring $(V,\mathfrak{m})$, and $K\langle t_1,\ldots, t_n\rangle$ a Tate algebra of convergent power series.
Fix $f \in V\langle t_1,\ldots, t_n\rangle$....

**8**

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322 views

### Why are the open and closed adic discs defined the way that they are?

The closed adic disc is defined as $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$, and the open adic disc is defined to be the fiber $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...

**9**

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293 views

### What role, if any, do Archimedean valuations play in adic spaces?

I've been reading about adic spaces, and I couldn't help but wonder what would happen to the theory if one included in the definition of $Spa$ Archimedean valuations as well...?
Is there a weird ...