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 (cf. Tate curve), 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 ![]() |
[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 ![]() ![]() ![]() |
[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 |
Tate algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_algebra&oldid=42381