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affinoid algebra

Context

Algebra

Analytic geometry

Contents

Idea

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

Definition

Let KK be a complete non-archimedean valued field.

As a ring, a standard affinoid algebra (or Tate algebra) T n,KT_{n,K} is the subring of the ring of [[formal power series in K[[x 1,,x n]]K[ [x_1, \ldots, x_n] ] consisting of all strictly converging series c= Ic Ix I c= \sum_I c_I x^I, that is such that |c I|0|c_I|\to 0 as II\to \infty.

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

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

The category of kk-affinoid spaces is the opposite category of the category of kk-affinoid algebras and bounded homomorphisms between them.

Properties

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

References

Affinoid algebras were introduced in

A standard textbook account is

See the references at analytic geometry for more details.

Discussion of affinoid algebras as a site for a more topos-theoretic formulation of of analytic geometry is in

See also

Revised on July 20, 2014 18:52:14 by David Roberts (129.127.211.140)