Questions tagged [rigid-analytic-spaces]
53 questions
8
votes
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(Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$
I am following Scholze's and Weinstein's notes on $p$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates).
$\operatorname{Spa}(\mathbb{Z},\mathbb{Z})$ is the space…
dyf
- 634
7
votes
1 answer
Definition of $p$-adic formal scheme
Could someone please provide a precise definition of a $p$-adic formal scheme $X$ over a ring $A$? Is it a formal scheme over $A$ which is locally isomorphic to $\operatorname{Spf}(B)$, where the completion is taken with respect to the ideal $(p)…
Legendre
- 949
7
votes
1 answer
What are Robba rings and why are they important?
If we study $p$-adic Galois representations we come across the so called "Robba rings", defined as the ring of Laurent series which converge in some annulus $\{z| \,\,R<|z|<1\}$ for some positive number $R<1$. I looked at the literature a bit to see…
Sameer Kulkarni
- 1,971
6
votes
3 answers
A question about the $p$-adic product formula for $\log(1+X)$ and $p$-adic geometry
Fix a prime number $p$. There is a well known product formula for $\log(1+X)$ as a power series in $\mathbb{Q}_p[[X]]$ (by Serre, I think). Namely letting $\Phi_n(X)=(1+X)^{p^n}-1$ and $Q_n(X)=\Phi_{n+1}(X)/p\Phi_n(X)$, then we…
xlord
- 285
6
votes
0 answers
Structure Sheaves of Rigid Analytic Spaces: What is the right Value Category?
Let $K$ be a complete non-archimedean field and let $X$ be an affinoid $K$-space. Then the structure sheaf $\mathcal O_X$ of $X$ is defined first on the "weak Grothendieck topology", ie on affinoid subdomains, and then by extending the sheaf to the…
benh
- 6,695
5
votes
1 answer
Canonical way to construct $R$ with $\mathrm{Spa}(A, A^+) \simeq \mathrm{Spec}(R)$?
In this Scholze-Weinstein's note on $p$-adic geometry, there's a Huber's theorem (Theorem 2.3.3): Adic spectrum is spectral, i.e. $\mathrm{Spa}(A, A^+)$ is homeomorphic to $\mathrm{Spec}( R)$ for some $R$ (which says that $\mathrm{Spa}$ is not that…
Seewoo Lee
- 15,670
4
votes
1 answer
Smooth affinoid rigid-analytic spaces over $\mathbb C_{p}$
Are there any examples of smooth affinoid rigid-analytic spaces $X$ over $\mathbb C_{p}$, which are not the base-change $X=Y_{\mathbb C_{p}}$
of a smooth rigid-analytic variety $Y$ over a finite extension of $\mathbb Q_{p}$?
user141099
- 250
4
votes
1 answer
Isomorphism of the perfection of two ring
I was working on Exercise 2.0.4 of Bhatt's notes, which are available here.
The exercise states:
Let $f\colon R\to S$ be a map of char $p$ rings that is surjective with nilpotent kernel. Then $R^{perf}$ and $S^{perf}$ are isomorphic. Here,
…
Chen
- 53
4
votes
0 answers
What is a ring $R$ s.t. $Spa(\mathbb{Z},\mathbb{Z})=Spec R$
The adic spectrum $Spa(\mathbb{Z},\mathbb{Z})$ looks as follows:
First, there is a point for every prime ideal $\mathfrak{p}\subset\mathbb{Z}$, corresponding to the valuation given by the composition of the quotient map $\mathbb{Z}\rightarrow…
jorst
- 1,615
3
votes
1 answer
What is the right notion of transcendence degree for the fraction field of an affinoid algebra
Let $A$ be a finitely generated $k$-algebra with fraction field $K$.
Then the Krull dimension of $A$ is equal to the transcendence degree of $K$ over $k$.
I would be very interested in any related result where $A$ is an affinoid algebra. I suppose…
Kemal
- 123
3
votes
0 answers
When is the zero set of a multivariate $p$-adic power series algebraic?
Let $f = f(z_1, \dots, z_n)$ be a power series in $n$ variables with coefficients in the $p$-adic integers $\mathbb{Z}_p$. Let $g(z_1, \dots, z_n) = f(pz_1, \dots, pz_n)$, so that $g$ converges on all of $\mathbb{Z}_p^n$. Under what conditions is…
Ashvin Swaminathan
- 2,836
3
votes
1 answer
Analytic functions on spaces over non-Archimedean fields and troubles with totally disconnectedness
I read in several intro scripts on Berkovish spaces that these
arose as new approach to analytic geometry
over non-archimedean fields. As the main problem in
non-archimedean analytic geometry is recognized the
observation that analytic…
user267839
- 9,217
3
votes
0 answers
Construct a subspace of a complete normed $K$-vector space with $(y_\mu)_{\mu \in M}$ as its orthonormal basis
I've been reading Bosch's book "Lectures on Formal and Rigid Geometry". In the proof of Theorem 6 on page 26, which I will show below, he claimed that there is a subspace $V'$ of a complete normed $K$-vector space $V$ with $(y_{\mu})_{\mu \in M}$ as…
ivy
- 31
3
votes
1 answer
"weak" Henselian property
I encountered following thread from MO treating the structure $X(k)$ of $k$-valued points of a separated algebraic space $X$ of finite type over $k$.
I have a question about an aspect from Laurent Moret-Bailly's amazing answer. let me quote:…
user705174
3
votes
1 answer
Nonarchimedean convergent power series
I would like to understand the second paragraph on the second page (marked page 320) of the article http://www.numdam.org/article/MSMF_1974__39-40__319_0.pdf on rigid analytic geometry by Michel Raynaud.
Let's take $K$ a complete nonarchimedean…
Mickey
- 720