Questions tagged [rigid-analytic-geometry]

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

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12
votes
1answer
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
0answers
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
1answer
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
votes
0answers
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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0answers
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
1answer
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
1answer
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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0answers
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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0answers
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
1answer
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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0answers
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
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0answers
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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0answers
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
0answers
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
votes
0answers
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
1answer
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
vote
0answers
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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0answers
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
1answer
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
1answer
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
vote
0answers
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
0answers
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
votes
0answers
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
votes
0answers
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
vote
0answers
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
votes
0answers
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
1answer
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
2answers
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
1answer
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
votes
0answers
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
vote
0answers
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
votes
1answer
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
votes
0answers
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
1answer
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
votes
0answers
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
votes
1answer
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
votes
1answer
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
1answer
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
1answer
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
1answer
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
vote
0answers
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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0answers
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
vote
0answers
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
votes
1answer
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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 ...