Affinoid Algebras

Abstract

An algebraic variety over a field k is obtained by glueing affine varieties overkwith 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.