nLab 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 non-archimedean valued 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 $K[ [x_1, \ldots, x_n] ]$ consisting of all strictly converging series $c= \sum_I c_I x^I$, that is such that $|c_I|\to 0$ as $I\to \infty$.

There is a Gauss norm? on such series $\|\sum_I c_I x^I \| = max\{|c_I|\}_I$. 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).

The category of $k$-affinoid spaces is the opposite category of the category of $k$-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: $T_{n,K}$ 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)

A standard textbook account is

• S. Bosch, U. Güntzer, Reinhold Remmert, part B of Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

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

• Oren Ben-Bassat, Kobi Kremnizer, section 7 of Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)

See also

• Federico Bambozzi, On a generalization of affinoid varieties (arXiv:1401.5702)