# Contents

## Idea

An affinoid algebra is a local model for analytic spaces in analytic geometry (rigid analytic geometry).

## Definition

Let $K$ be a complete ultrametric field?.

As a ring, a standard affinoid algebra (or Tate algebra) $T_{n,K}$ is the subring of the ring of [[formal power series in $K[ [x_1, \ldots, x_n] ]$ consisting of all strictly converging series $c= \sum_I c_I x^I$, that is such that $|c_I|\to 0$ as $I\to \infty$.

There is a Gauss norm? on such series $\|\sum_I c_I x^I \| = max\{|c_I|\}_I$. This is indeed a norm making $T_{n,K}$ into a Banach $K$-algebra of countable type.

An affinoid algebra is any Banach algebra which can be represented in a form (Tate algebra)/(closed ideal).

## Properties

A version of the Weierstrass preparation theorem in this context implies a version of the Hilbert basis theorem: $T_{n,K}$ is a noetherian ring. Moreover $T_{n,K}$ is a unique factorization domain of Krull dimension? $n$.

## References

Affinoid algebras were introduced in

See the references at analytic geometry for more details.

Revised on January 6, 2012 11:43:50 by Urs Schreiber (89.204.137.240)