Math 2.0

Algebraic geometry Definition

Rigid analytic space

Builds on Non-Archimedean absolute valuep-adic numberRinged spaceSheaf

Definition

Let KK be a field complete for a nontrivial non-archimedean absolute value ∣⋅∣\abs{\cdot}. The Tate algebra in nn variables is Tn=K⟨x1,…,xn⟩={∑ν∈ℕnaνxν:aν∈K, ∣aν∣→0 as ∣ν∣→∞},T_n = K\langle x_1, \dots, x_n \rangle = \set{\textstyle\sum_{\nu \in \N^n} a_\nu x^\nu : a_\nu \in K,\ \abs{a_\nu} \to 0 \text{ as } \abs{\nu} \to \infty}, the algebra of power series that converge on the closed unit polydisc. An affinoid algebra is a KK-algebra AA isomorphic to a quotient Tn/IT_n / I for some nn and some ideal II. The affinoid space Sp⁡A\operatorname{Sp} A is the set of maximal ideals of AA with a Grothendieck topology and a structure sheaf 𝒪\mathcal{O}. In this topology the admissible opens are the finite unions of affinoid subdomains. The admissible coverings are the finite coverings by such sets. A rigid analytic space over KK is a set XX with a Grothendieck topology and a sheaf of KK-algebras, as a ringed space. It must have an admissible covering by open subsets each isomorphic, with these structures, to an affinoid space.

Remarks

That 𝒪\mathcal{O} is a sheaf for finite affinoid coverings is Tate's acyclicity theorem. The general admissible opens and coverings are defined from these by a standard extension of the topology.

Example: Sp⁡T1\operatorname{Sp} T_1 is the closed unit disc. Its points are the maximal ideals of T1T_1, which correspond to elements xx of an algebraic closure of KK with ∣x∣≤1\abs{x} \le 1, up to conjugation over KK.

Non-example of an admissible covering: for K=ℚpK = \Q_p, the closed disc Sp⁡T1\operatorname{Sp} T_1 is the disjoint union of the open disc {∣x∣<1}\set{\abs{x} < 1} and the circle {∣x∣=1}\set{\abs{x} = 1}. This covering is not admissible. Every affinoid subdomain of the open disc lies in a disc {∣x∣≤r}\set{\abs{x} \le r} with r<1r < 1. So no finite family of affinoid subdomains of the two pieces covers the points with r<∣x∣<1r < \abs{x} < 1. This is what keeps the closed disc connected in rigid geometry.

The analytification of a variety of finite type over KK is a rigid analytic space, and a rigid GAGA theorem compares coherent sheaves on both sides for proper varieties.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Non-Archimedean absolute value, concept, no evidence levelNon-Archimedean absol…conceptRinged space, concept, no evidence levelRinged spaceconceptSheaf, concept, no evidence levelSheafconceptp-adic number, concept, no evidence levelp-adic numberconceptRigid analytic space, definition, no evidence levelRigid analytic spacedefinition
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  1. Non-Archimedean absolute value concept
  2. Ringed space concept
  3. Sheaf concept
  4. p-adic number concept
  5. Rigid analytic space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rigid_analytic_space
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Kinddefinition
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MSC14G22, 12J25
DefinesRigid analytic space
UsesNon-Archimedean absolute value, p-adic number, Ringed space, Sheaf

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