Mathematics

Questions tagged [rigid-analytic-spaces]

59 questions
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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
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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
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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…
Sameer Kulkarni
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6
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Generic and Special Fibres for Rigid Analytic Model $\Bbb G_{m}^{\text{an}}$ of Multiplicative Group

An absolute beginner level question on rigid analytic spaces & their interplay with formal schemes (after Raynaud). In this minicourse by Bosch on this topic he introduced as motivation following pictures whose meaning I not completely understand,…
user267839
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6
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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 have…
xlord
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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
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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
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What is the complement of the closed unit disk in the projective line

Let $K$ be a nonarchimedean field, and let $\mathbf{P}^1_K$ be the projective line over $K$ , always viewed as an adic space. Let $D$ be the closed unit disk embedded into $\mathbf{P}^1_K$ in the standard way; so $D$ is an affinoid open in…
user393795
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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
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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
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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
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What is the (p-adic) ghost circle in Colmez--Gilles--Nizioł

All spaces in this question are adic spaces. In the paper Pierre Colmez, Sally Gilles, and Wiesława Nizioł. Arithmetic duality for p-adic pro-étale cohomology of analytic curves. arXiv:2308.07712, 2023. the authors define a curious object called the…
user393795
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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…
Kemal
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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…
Ashvin Swaminathan
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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 functions…
user267839
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