Author: Eiko

Time: 2025-04-07 08:30:08 - 2025-04-07 08:30:08 (UTC)

Affinoid Algebras

Intuition: Some analogies

Algebraic and Rigid Spaces

Algebraic variety = affine varieties glued by Zariski topology.

Rigid space = affinoid spaces glued by G-topology (a type of Grothendieck topology).

Affinoid Algebras and Holomorphic Functions

Complex holomorphic functions are defined on some subset of ${\mathbb{C}}^n$.

Affinoid algebras are functions on some subsets of $k^n$.

This viewpoint is especially valid if $k$ is algebraically closed.

Definition

The standard affinoid algebra, also called the standard Tate algebra, $T_{n,k}=k⟨z_1,\dots ,z_n⟩$ is the sub-ring of formal power series $k[[z_1,\dots ,z_n]]$ converging on the closed unit ball $D^n(0,1)$, i.e. $|c_{\alpha }|\to 0$ as $\alpha \to \mathrm{\infty }$.

There is a usual Gauss norm $\parallel \sum c_{\alpha }{z}^{\alpha }\parallel =\underset{\alpha }{sup}|c_{\alpha }|=\underset{\alpha }{max}|c_{\alpha }|$.

• Write $T_n^{\mathfrak{o}}=\{\parallel f\parallel \le 1\}$, $T_n^{\mathfrak{m}}=\{\parallel f\parallel <1\}$.

• We have the following basic properties:

  • $T_n$ is a complete normed space, a Banach algebra.

  • $T_n$ is a $k$-algebra

  • $T_n^{\mathfrak{o}}$ is a $k^{\mathfrak{o}}$-algebra

  • $T_n^{\mathfrak{m}}\subset T_n^{\mathfrak{o}}$ is an ideal

  • ${\bar{T}}_n:=T_n^{\mathfrak{o}}/T_n^{\mathfrak{m}}\cong \bar{k}[z_1,\dots ,z_n]$ is a $\bar{k}$-algebra

An affinoid algebra or Tate algebra $A$ over $k$ is a $k$-algebra which is a finite extension of some $T_{n,k}$, i.e. there exists a finite $k$-algebra homomorphism $T_n\to A$

The Tate algebra shares many properties with the familiar polynomial ring ${\bar{T}}_n=\bar{k}[z_1,\dots ,z_n]$. A powerful tool for affinoid algebras is the Weierstrass theorem

Theorem (Weierstrass Preparation and Division)

• (Division) We can perform residue division from a regular series. Let $f\in T_n$ be regular in $z_n$ of degree $d$, i.e. $\bar{f}=\lambda z_n^d+\sum c_i(z_1,\dots ,z_{n-1})z_n^i$ with $\lambda \ne 0\in \bar{\lambda }$ and $c_i\in {\bar{T}}_{n-1}$.

  Then there exists residue division algorithm in $T_n$ respect to $f$, for any $g\in T_n$ there is $q\in T_n$ and $r\in T_{n-1}[z_n]$ with ${\mathrm{deg}}_{z_n}r<d$ such that

  $$g=qf+r,\parallel g\parallel =max(\parallel q\parallel ,\parallel r\parallel ).$$

• (Preparation) We can put elements in its regular form. If $f\in T_n$ has norm $1$, there exists a $k$-algebra automorphism $\sigma$ of $T_n$ such that $\sigma (f)$ is regular in $z_n$.

Consequences

• Division implies that the quotient

  $$T_n/(f){\cong }_{T_{n-1}}\underset{i=0}{\overset{d-1}{⨁}}{T}_{n-1}{z}_n^i$$

  is a free $T_{n-1}$-module of rank $d$.

• It also implies that for any ideal $I\subset T_n$, after some linear change of variables we can choose a regular $f\in I$ such that

  $$I=⟨f,I\cap T_{n-1}[z_n]⟩$$

  where $I\cap T_{n-1}[z_n]$ is an ideal of $T_{n-1}[z_n]$.

• Any affinoid algebra is Noetherian.

  Proof. Let $I$ contain a regular $f$, any $g\in I$ writes as $g=qf+r$ with $r\in T_{n-1}[z_n]$, so $I=⟨f,I\cap T_{n-1}[z_n]⟩$, the latter ring is Noetherian by induction and so the ideal $I\cap T_{n-1}[z_n]$ is finitely generated.

• $T_n$ is UFD with Krull dimension $n$.

• $\parallel \cdot \parallel$ be any norm on affinoid algebra $A$ making $A$ into a Banach $k$-algebra, then every ideal of $A$ is closed with respect to this norm.

• (Normalization) For ideal $I\subset T_n$ there exists an integer $d$ and an injective finite map $T_d\to T_n/I$ (a finite $T_d$-module). Geometrically this means $I$ defines a $d$-dimensional space, lying finitely over $T_d$.

• (Nullstellensatz) For any maximal ideal $\mathfrak{m}\subset T_n$, $T_n/\mathfrak{m}$ is a finite extension of $k$.

• Every affinoid algebra is of the form $T_n/I$, the Gauss-norm induces a norm on $A$ making it a Banach algebra.

• A morphism of Banach algebras $u:(A_1,\parallel \cdot {\parallel }_1)\to (A_2,\parallel \cdot {\parallel }_2)$ is continuous with respect to the norms.

• All norms on an affinoid algebra making Banach algebra, are equivalent.

Max Spectrum

If $A$ is an affinoid algebra, we denote by $\mathrm{Sp}(A)$ the maximal spectrum of $A$, i.e. the set of all maximal ideals of $A$. Note that it has Zariski topology and totally discontinuous topology, but they are not for rigid geometry.