An affinoid algebra is a local model for analytic spaces in analytic geometry (rigid analytic geometry).
Let be a complete ultrametric field?.
As a ring, a standard affinoid algebra (or Tate algebra) is the subring of the ring of [[formal power series in consisting of all strictly converging series , that is such that as .
There is a Gauss norm? on such series . This is indeed a norm making into a Banach -algebra of countable type.
An affinoid algebra is any Banach algebra which can be represented in a form (Tate algebra)/(closed ideal).
A version of the Weierstrass preparation theorem in this context implies a version of the Hilbert basis theorem: is a noetherian ring. Moreover is a unique factorization domain of Krull dimension? .
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