symmetric monoidal (∞,1)-category of spectra
An affinoid algebra is a local model for analytic spaces in analytic geometry (rigid analytic geometry).
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.
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 $n$.
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
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