Algebras and modules
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Geometry on formal duals of algebras
An affinoid algebra is a local model for analytic spaces in analytic geometry (rigid analytic geometry).
Let be a complete non-archimedean valued 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).
The category of -affinoid spaces is the opposite category of the category of -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: is a noetherian ring. Moreover is a unique factorization domain of Krull dimension? .
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
Revised on July 20, 2014 18:52:14
by David Roberts