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affinoid algebra

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 $K[[x_1,\dots ,x_n]]$ consisting of all strictly converging series $c={\sum }_I{c}_I{x}^I$, that is such that $\mid c_I\mid \to 0$ as $I\to \mathrm{\infty }$.

There is a Gauss norm? on such series $\parallel {\sum }_I{c}_I{x}^I\parallel =\max \{\mid c_I\mid {\}}_I$. 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

• John Tate, (1961)

See the references at analytic geometry for more details.