Tate algebra

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, Mumford curves and surfaces, Shimura curves and varieties, etc.

References

[a1]	S. Bosch, U. Güntzer, R. Remmert, "Non-Archimedean analysis" , Springer (1984)
[a2]	V.G. Drinfel'd, "Coverings of -adic symmetric regions" Funct. Anal. Appl. , 10 : 2 (1976) pp. 107–115 Funkts. Anal. Prilozhen. , 10 : 2 pp. 29–41
[a3]	G. Faltings, "Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten" , Lect. notes in math. , 1111 , Springer (1984)
[a4]	J. Fresnel, M. van der Put, "Géométrie analytique rigide et applications" , Birkhäuser (1981)
[a5]	L. Gerritzen, M. van der Put, "Schottky groups and Mumford curves" , Lect. notes in math. , 817 , Springer (1980)
[a6]	D. Mumford, "An analytic construction of degenerating curves over complete local fields" Compos. Math. , 24 (1972) pp. 129–174
[a7]	D. Mumford, "An analytic construction of degenerating abelian varieties over complete rings" Compos. Math. , 24 (1972) pp. 239–272
[a8]	D. Mumford, "An algebraic surface with ample, , " Amer. J. Math. , 101 (1979) pp. 233–244
[a9]	M. Raynaud, "Variétés abéliennes en géométrie rigide" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 2 , Gauthier-Villars (1971) pp. 473–477
[a10]	J. Tate, "Rigid analytic spaces" Invent. Math. , 12 (1971) pp. 257–289