Tate algebra

Summary

The Tate algebra in $n$ variables over a complete non-archimedean field $K$ is the algebra of power series whose coefficients tend to $0$. These are exactly the power series that converge on the closed unit polydisc over an algebraic closure of $K$. With the Gauss norm it is a Banach algebra, and it is a Noetherian unique factorization domain. Its quotients, the affinoid algebras, are the local building blocks of rigid analytic spaces, as polynomial rings are for algebraic varieties.