Algebraic variety = affine varieties glued by Zariski topology.
Rigid space = affinoid spaces glued by G-topology (a type of Grothendieck topology).
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.
The standard affinoid algebra, also called the standard Tate algebra, \(T_{n,k}=k\langle z_1,\dots, z_n\rangle\) 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 \(\alpha \to \infty\).
There is a usual Gauss norm \(\|\sum c_\alpha z^\alpha\| = \sup_\alpha |c_\alpha| = \max_\alpha |c_\alpha|\).
Write \(T_n^\mathfrak{o}=\{\|f\|\le 1\}\), \(T_n^\mathfrak{m}=\{\|f\|<1\}\).
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
\(\overline{T}_n:= T_n^\mathfrak{o}/T_n^\mathfrak{m}\cong \overline{k}[z_1,\dots,z_n]\) is a \(\overline{k}\)-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 \(T_n\rightarrow A\)
The Tate algebra shares many properties with the familiar polynomial ring \(\overline{T}_n = \overline{k}[z_1,\dots,z_n]\). A powerful tool for affinoid algebras is the Weierstrass theorem
(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. \(\overline{f} = \lambda z_n^d+\sum c_i(z_1,\dots,z_{n-1})z_n^i\) with \(\lambda \neq 0 \in \overline{\lambda}\) and \(c_i\in \overline{T}_{n-1}\).
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 \(\deg_{z_n} r < d\) such that
\[g = qf + r, \quad \|g\| = \max(\|q\|, \|r\|).\]
(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 \(z_n\).
Division implies that the quotient
\[T_n/(f)\cong_{T_{n-1}} \bigoplus_{i=0}^{d-1} T_{n-1}z_n^i\]
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
\[I = \langle f, I\cap T_{n-1}[z_n]\rangle\]
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 \(I = \langle f, I\cap T_{n-1}[z_n]\rangle\), 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 \(n\).
\(\|\cdot\|\) 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 \(u:(A_1,\|\cdot\|_1) \to (A_2, \|\cdot\|_2)\) is continuous with respect to the norms.
All norms on an affinoid algebra making Banach algebra, are equivalent.
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.