nLab affinoid algebra




Analytic geometry



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


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.


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.


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

Last revised on November 24, 2018 at 17:09:14. See the history of this page for a list of all contributions to it.