# Tate algebra

(Redirected from Affinoid algebra)
Jump to: navigation, search

Let be a field which is complete with respect to an ultrametric valuation (i.e. ). The valuation ring has a unique maximal ideal, . The field is called the residue field of .

Examples of such fields are the local fields, i.e. finite extensions of the -adic number field , or the field of Laurent series in with coefficients in the finite field (cf. also Local field).

Let denote indeterminates. Then denotes the algebra of all power series with ( ) such that ( ). The norm on is given by . The ring is denoted by , and is an ideal of . Then is easily seen to be the ring of polynomials .

The -algebra is called the free Tate algebra. An affinoid algebra, or Tate algebra, over is a finite extension of some (i.e. there is a homomorphism of -algebras which makes into a finitely-generated -module). The space of all maximal ideals, of a Tate algebra is called an affinoid space.

A rigid analytic space over is obtained by glueing affinoid spaces. Every algebraic variety over has a unique structure as a rigid analytic space. Rigid analytic spaces and affinoid algebras were introduced by J. Tate in order to study degenerations of curves and Abelian varieties over .

The theory of formal schemes over (the valuation ring of ) is close to that of rigid analytic spaces. This can be seen as follows.

Fix an element with . The completion of with respect to the topology given by the ideals is the ring of strict power series over . Now , and is the localization of with respect to . So one can view as the "general fibre" of the formal scheme over . More generally, any formal scheme over gives rise to a rigid analytic space over , the "general fibre" of . Non-isomorphic formal schemes over can have the same associated rigid analytic space over . Further, any reasonable rigid analytic space over is associated to some formal scheme over .

Affinoid spaces and affinoid algebras have many properties in common with affine spaces and affine rings over . Some of the most important are: Weierstrass preparation and division holds for (cf. also Weierstrass theorem); affinoid algebras are Noetherian rings, and even excellent rings if the field is perfect; for any maximal ideal of an affinoid algebra the quotient field is a finite extension of ; many finiteness theorems; any coherent sheaf on an affinoid space is associated to a finitely-generated -module (further: for ).

Another interpretation of is: consists of all "holomorphic functions" on the polydisc . This interpretation is useful for finding the holomorphic functions on more complicated spaces like Drinfel'd's symmetric spaces . Let be a local field with algebraic closure . Then  is a Drinfel'd symmetric space.

Spaces of this type have been used for the construction of Tate's elliptic curves (cf. Tate curve), Mumford curves and surfaces, Shimura curves and varieties, etc.