All Questions
Tagged with rigid-analytic-geometry ag.algebraic-geometry
128 questions
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 $X$? ...
- 643
7
votes
0
answers
416
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 ...
- 4,418
3
votes
1
answer
275
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 ...
- 1,093
11
votes
0
answers
375
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 ...
- 16.2k
1
vote
0
answers
190
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 ...
- 949
3
votes
1
answer
307
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^{...
- 452
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 ...
- 228
2
votes
0
answers
156
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 ...
- 6,622
4
votes
0
answers
281
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],\...
- 1,093
7
votes
0
answers
257
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 ...
- 13.1k
4
votes
1
answer
393
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 ...
- 1,838
12
votes
1
answer
510
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" ${\...
- 16.2k
6
votes
0
answers
356
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 ...
- 3,539
2
votes
1
answer
381
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 ...
- 217
5
votes
1
answer
506
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 ...
- 11.8k
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 ...
- 11.8k
8
votes
1
answer
384
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 ...
- 1,153
6
votes
1
answer
672
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. ...
- 1,153
6
votes
1
answer
670
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 (...
- 2,789
4
votes
1
answer
225
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 ...
- 7,746
8
votes
0
answers
550
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 ...
- 2,923
3
votes
1
answer
403
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 ...
user138661
8
votes
0
answers
354
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 ...
user138661
4
votes
1
answer
279
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 $p$-...
user138661
7
votes
1
answer
480
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 ...
- 37k
8
votes
1
answer
536
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 ...
user137712
7
votes
1
answer
765
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 $...
- 2,923
2
votes
0
answers
127
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 $k$...
- 257
4
votes
0
answers
307
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 ...
10
votes
1
answer
564
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-...
- 435
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 $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
- 64k
10
votes
0
answers
269
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 $f \in V\langle t_1,\ldots, t_n\rangle$....
user129156
9
votes
0
answers
687
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 $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
- 2,081
9
votes
0
answers
327
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 ...
- 2,081
4
votes
1
answer
464
views
Tropical charts (coordinates) and differential forms in non-archimedean geometry
Chambert-Loir and Ducros have introduced real differential forms and currents on Berkovich spaces.(See Gubler's survey for example). In that survey, a tropical chart $V$ is defined on an ...
- 2,789
8
votes
1
answer
1k
views
Reference Request: Specialization map in Huber's Context
The specialization map $sp:\mathfrak{X}_\eta\to \mathfrak{X}_{red}$ has an important role in rigid analytic geometry. I tried looking in Huber's papers ("Continuous Valuations", "A generalization of ...
- 435
23
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 ...
- 23.4k
5
votes
0
answers
1k
views
Formal GAGA and étale cohomology
Let $\mathfrak{X}$ be a $p$-adic flat formal scheme over $\mathbf{Z}_p$, whose special fiber has an ample line bundle. Then $\mathfrak{X}$ is algebraizable, that is there exists an algebraic $\mathbf{...
user92332
3
votes
0
answers
279
views
Analytic Space with No Regular Points
Define an analytic space to be a topological space $X$ equipped with a sheaf of rings $\mathcal{O}_X$ such that for every point $x \in X$ there is a neighbourhood $U \subseteq X$ such that $(U, \...
- 39
32
votes
1
answer
8k
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 ...
user113453
11
votes
3
answers
2k
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 $...
user113393
3
votes
0
answers
275
views
Seminorms on tensor products of affinoid algebras
Let $A \leftarrow C \rightarrow B$ be affinoid $K$-algebras, where $K$ is a non-archimedean field with non-trivial absolute value. Equipping $A$, $B$, $C$ with the supremum seminorms, there is a ...
- 31
3
votes
0
answers
331
views
Etale cohomology of rigidification
Let $K/\mathbb Q_p$ be a discretely valued non-archemedean field, let $X$ be a smooth scheme over $\mathcal O_K$. To $X$ one can associate two rigid-analytic spaces over $K$:
1) the analytification $...
- 790
8
votes
0
answers
310
views
Disconnectedness in Rigid Analytic Geometry: a technical question
I have a technical question in rigid analytic geometry, in the sense of Tate.
Let $B$ be an affinoid space over $\mathbb Q_p$, $Z$ a rigid analytic variety with a flat surjective map $f:Z \...
- 26.1k
1
vote
0
answers
165
views
Rigid analytic reductions of the projective line
I'm reading the book "Rigid Analytic Geometry and its Applications" by Fresnel-van der Put, and I'm confused by their example 4.8.5. In the first two parts of the example, they define the analytic ...
- 164
2
votes
1
answer
329
views
Representation of elements in affinoid algebra
Let $K$ be a complete, algebraically closed non-Archimedean field, and let $p \in K[x]$ be of degree $d > 0$ and norm 1. (Here the norm of a polynomial is the maximum of the norms of its ...
2
votes
0
answers
115
views
Converging sequence of base change
Here is a natural question that I hope will be of interest to some.
Let $\mathbf{F}_p(\!(T)\!)$ be the field of formal Laurent series over $\mathbf{F}_p$. An automorphism of $\mathbf{F}_p(\!(T)\!)$ ...
- 1,017
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 (...
- 393
4
votes
1
answer
421
views
Representability of relative Hilbert and Picard functors over analytic spaces
Let $f:X \to S$ be a morphism of complex analytic spaces. Then, just like in the case of schemes, we can define the relative Hilbert and Picard functors. For instance, if $\text{An}_{/S}$ denotes de ...
user96873
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 ...
- 843