# Affinoid Algebras

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## Abstract

An algebraic variety over a field *k* is obtained by glueing affine varieties over*k*with respect to the Zariski topology. Further, an affine variety is the set of maximal ideals of some finitely generated algebra over *k*. Rigid (analytic) spaces over a complete non-archimedean valued field *k* are formed in a similar way. A rigid space is obtained by glueing affinoid spaces with respect to a certain Grothendieck topology which we will call a *G*-topology. An affinoid space is the set of maximal ideals of certain *k*-algebras called affinoid algebras. One can think of an affinoid algebra over *k* as an algebra of functions defined on suitable subsets of *k*^{ d }, in analogy with complex holomorphic functions defined on open subsets of C^{ d }. This point of view is especially valid if the field *k* is algebraically closed. Alternatively, an affinoid algebra over *k* can be thought of as a completion of a finitely generated *k*-algebra. Indeed, affinoid algebras share many properties with finitely generated *k*-algebras. However, the definitions and proofs in the world of affinoid algebras are often more technical than the corresponding features for finitely generated *k*-algebras. We hope that the examples of the text will clarify some of the technical details. We note that the prime spectrum, i.e., Spec(*A*), of an affinoid algebra *A* is less relevant to the theory of rigid spaces because its usual Zariski topology cannot be used for glueing affinoid spaces.

## Keywords

Prime Ideal Maximal Ideal Banach Algebra Spectral Norm Minimal Prime Ideal## Preview

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