Questions tagged [rigid-analytic-geometry]

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

Filter by
Sorted by
Tagged with
44 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-...
Wojowu's user avatar
  • 27.4k
41 votes
2 answers
3k views

Perfectoid universal covers

It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this ...
SomeGuy's user avatar
  • 833
39 votes
1 answer
13k views

Why is Faltings' "almost purity theorem" a purity theorem?

My understanding of purity theorems is that they come in several flavors: 1) Those of the form "this Galois representation is pure, i.e. the eigenvalues of $Frob_p$ are algebraic numbers all of whose ...
user34143's user avatar
  • 575
35 votes
0 answers
1k 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 $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
Tim Campion's user avatar
  • 60.6k
32 votes
1 answer
7k views

$p$-adic Hodge Theory for rigid spaces, after P. Scholze

I was going over P. Scholze's paper on $p$-adic Hodge Theory for rigid analytic varieties. This question is around the "Poincaré Lemma" in the paper. Throughout, let $X$ be a proper smooth rigid ...
user avatar
31 votes
1 answer
2k views

Structure on $X(k)$ for separated finite type alg. space $X$, for complete valued $k$.

Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$. If $X$ is a scheme then $X(k)$ inherits a natural (...
27 votes
1 answer
1k 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 ...
Emily's user avatar
  • 10.3k
26 votes
3 answers
7k views

A roadmap for understanding perfectoid spaces

Perfectoid spaces are this year's subject for the Arizona Winter School (link) and, as preparation, I am currently trying to understand the subject better. There are wonderful explanatory accounts (...
user8329099's user avatar
24 votes
2 answers
4k views

Rigid analytic spaces vs Berkovich spaces vs Formal schemes

I wonder if someone could explain briefly what is the relation between these 3 formal models, of a Berkovich space, a rigid analytic space and a formal scheme? I have been working with formal schemes ...
geometer's user avatar
  • 241
23 votes
1 answer
1k views

function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
Urs Schreiber's user avatar
21 votes
3 answers
1k views

Intuition about $\mathrm{Spec}\mathbb{C}[[t]]$ versus $\mathrm{Spf}\mathbb{C}[[t]]$ versus $\mathrm{Specan}\mathbb{C}[[t]]$ (and similar objects)

The first one $\mathrm{Spec}\mathbb{C}[[t]]$ is a scheme, the second one $\mathrm{Spf}\mathbb{C}[[t]]$ is a formal scheme. In my mind they both realize an "infinite order infinitesimal neighbourhood ...
Qfwfq's user avatar
  • 22.7k
20 votes
3 answers
2k views

Étale homotopy type of non-archimedean analytic spaces

The following is likely all obvious to the experts. But since the field looks tricky to an outsider, maybe I may be excused for asking anyway. I am wondering about basic facts of what would naturally ...
Urs Schreiber's user avatar
19 votes
1 answer
2k views

Are flat morphisms of analytic spaces open?

Let $f:X\to Y$ be a morphism of complex analytic spaces. Assume $f$ is flat (or, more generally, that there is a coherent sheaf on $X$ with support $X$ which is $f$-flat). Is $f$ an open map? The ...
Laurent Moret-Bailly's user avatar
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 ...
Marsault Chabat's user avatar
18 votes
1 answer
1k views

$p$-adic Bott periodicity?

The Bott periodicity theorem can be formulated as the existence of homotopy equivalences $\Omega^2(KU)\equiv KU$ and $\Omega^8(KO)=KO$. I always wondered whether this theorem could also be transferred ...
Dominik's user avatar
  • 3,007
18 votes
1 answer
1k views

Why do rigid spaces have "not enough points"?

In Brian Conrad's notes here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero ...
user34143's user avatar
  • 575
17 votes
0 answers
930 views

A functor of points approach to Berkovich analytic spaces

Is it possible to define a Berkovich analytic space via its functor of points? Let $k$ be a complete non-Archimedean field, possibly the trivial one. I am tempted to define a Berkovich analytic space ...
Martin Ulirsch's user avatar
16 votes
1 answer
1k views

D-modules on rigid analytic spaces

Is there a good notion of holonomic $D$-modules on rigid analytic spaces?
Anonymous's user avatar
  • 491
14 votes
1 answer
816 views

How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$?

I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties. In the ...
Keenan Kidwell's user avatar
14 votes
0 answers
540 views

Vanishing of rigid cohomology for affine varieties

Let $k$ be a perfect field of positive characteristic and denote by $K$ the field of fractions of the ring of Witt vectors over $k$. Question: If $X$ is an affine variety over $k$, do the rigid ...
Niels's user avatar
  • 141
13 votes
2 answers
2k views

Cohomology of rigid-analytic spaces

Let $R$ be a complete discrete valuation ring and let $K$ be its field of fractions. Suppose $X$ is a smooth rigid-anaytic space over $K$. Often it is convenient to have a model of $X$ whose ...
Jared Weinstein's user avatar
13 votes
1 answer
2k views

Reference for rigid analytic GAGA

I'm looking for a reference for the following result. Theorem. Let $K$ be a complete, non-archimedean field, and let $X/K$ be a projective scheme, with analytification $X^\mathrm{an}$. Then the ...
ChrisLazda's user avatar
  • 1,818
12 votes
1 answer
484 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 ...
Ashwin Iyengar's user avatar
12 votes
1 answer
466 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" ${\...
Piotr Achinger's user avatar
12 votes
1 answer
2k views

(pro)Étale cohomology of adic spaces and inverse limit

I am studying Peter Scholze's paper $p$-adic Hodge theory for rigid-analytic varieties and I am confused by the following. Let $X$ be a finite type scheme over $\mathbb{C}_p$ (proper and smooth if ...
franck's user avatar
  • 273
11 votes
3 answers
1k views

Trivialisation of vector bundles on Stein spaces

Does every vector bundle on a Stein space have a finite local trivialisation? Definitions: Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...
Helene Sigloch's user avatar
11 votes
1 answer
787 views

Consequences of the geometric properties of the eigencurve

The eigencurve $\mathcal{E}$ is a rigid-analytic space parametrizing certain $p$-adic families of modular forms and associated Galois representations. By constructing an auxiliary reduced rigid curve ...
user12235's user avatar
  • 163
11 votes
1 answer
825 views

bornological vector spaces over a non-archimedean field

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in '...
Oren Ben-Bassat's user avatar
11 votes
1 answer
1k views

Do Berkovich homogenous spaces exist?

Let G be a k-analytic group, and let H be a closed subgroup of G. Then does there exist a k-analytic space, which can be reasonably called the quotient G/H? Commentary: I realise that I am not being ...
Peter McNamara's user avatar
11 votes
3 answers
1k views

p-adic Poincaré Lemma

suppose $X$ is a proper and smooth rigid analytic variety over $\text{Spa}(k)$, with $k$ a non-archimedean field of characteristic zero. One has the de Rham complex of analytic differential forms on $...
user avatar
11 votes
0 answers
333 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 ...
Piotr Achinger's user avatar
11 votes
0 answers
371 views

Symplectic structures in rigid geometry

Let $K$ be a non-archimedean valued field (with any further adjectives attached as necessary). I'm looking for references or information about symplectic structures on rigid $K$-spaces. For example, ...
Mr. Chip's user avatar
  • 169
11 votes
0 answers
454 views

Is the Gouvea-Mazur problem related to symmetric square $L$-functions?

Here's an idea that I've found appealing but have never been able to get anywhere with. One way to frame the Gouvea-Mazur question (for lack of a better term, since the original conjecture by the ...
Ramsey's user avatar
  • 2,763
10 votes
2 answers
816 views

Uniqueness of analytic continuation in rigid analytic geometry

In classical complex analysis it is easy to prove that a meromorphic function has at most one analytic continuation (on an open connected subset of $\mathbb C$, say). The problem of non-uniqueness of ...
John's user avatar
  • 121
10 votes
1 answer
1k views

Analytic elements in non-archimedean geometry

Let $(k,|.|)$ be a complete non-archimedean valued field. Let $D$ be the open unit disc over $k$. (Anything I write could be adapted to the case of an open annulus.) The ring $\mathcal{O}(D)$ of ...
Jérôme Poineau's user avatar
10 votes
1 answer
489 views

Picard group of Drinfeld upper half space

Let $K$ be a $p$-adic field and $\Omega^{(n)}_K$ the $n$-dimensional Drinfeld upper half space over $K$ (which is a rigid analytic space over $K$). Is the Picard group of $\Omega^{(n)}_K$ known? ...
naf's user avatar
  • 10.5k
10 votes
0 answers
259 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 $f \in V\langle t_1,\ldots, t_n\rangle$....
user avatar
10 votes
0 answers
391 views

Detecting $k$-affinoid spaces by vanishing cohomology

The property of being an affine scheme can be tested against all quasi-coherent sheaves in the following sense: a noetherian scheme $X$ is affine iff $H^i(X,\mathcal{F}) = 0$ for all quasi-coherent $\...
msteve's user avatar
  • 572
9 votes
1 answer
1k 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}...
Jon Aycock's user avatar
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 ...
RumDiary's user avatar
  • 228
9 votes
1 answer
573 views

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

Let $A = \mathcal{C}(X)$ be a commutative (unital) C*-Algebra. Let $Spec(A)$ denote its Gelfand spectrum $$ Spec(A) = \{A \rightarrow \mathbb{C} : \text{non-zero *-homomorphism} \} \simeq X. $$ Now ...
Niki's user avatar
  • 335
9 votes
1 answer
529 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-...
Ian Gleason's user avatar
9 votes
0 answers
379 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 ...
pupshaw's user avatar
  • 848
9 votes
0 answers
613 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 $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
Andrew NC's user avatar
  • 2,011
9 votes
0 answers
321 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 ...
Andrew NC's user avatar
  • 2,011
9 votes
0 answers
261 views

Kodaira embedding theorem for rigid analytic varieties

Kodaira embedding theorem can be regarded as a vast generalizaton of the projectivity criterion for complex tori: indeed, the Riemann conditions essentially say that the line bundle defined by the ...
Dima Sustretov's user avatar
8 votes
3 answers
524 views

Weierstrass points on rigid-analytic surfaces

Does a rigid-analytic surface defined over a nonarchimedean complete field have Weierstrass points (if its genus is big enough let's say)? Is there a good reference that (ideally) lists theorems for ...
C Vincent's user avatar
8 votes
1 answer
311 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}...
Fernando Peña Vázquez's user avatar
8 votes
1 answer
704 views

What are the automorphisms of a perfectoid Tate algebra?

Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a_1T+a_2T^2+\cdots$, where $a_1\in \mathcal{O}...
Jared Weinstein's user avatar
8 votes
1 answer
394 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 \...
Dat Minh Ha's user avatar
  • 1,472

1
2 3 4 5