Tate algebra
Let
be a field which is complete with respect to an ultrametric valuation | \cdot | (
i.e. | x+ y | \leq \max ( | x | , | y | ) ).
The valuation ring R= \{ {a \in K } : {| a | \leq 1 } \}
has a unique maximal ideal, m= \{ {a \in K } : {| a | < 1 } \} .
The field k= R/m
is called the residue field of K .
Examples of such fields are the local fields, i.e. finite extensions of the p - adic number field \mathbf Q _ {p} , or the field of Laurent series \mathbf F _ {p} (( t)) in t with coefficients in the finite field \mathbf F _ {p} = \mathbf Z / p \mathbf Z ( cf. also Local field).
Let z _ {1} \dots z _ {n} denote indeterminates. Then T _ {n} ( K) = K \langle z _ {1} \dots z _ {n} \rangle denotes the algebra of all power series \sum a _ \alpha z _ {1} ^ {\alpha _ {1} } \dots z _ {n} ^ {\alpha _ {n} } with a _ \alpha \in K ( \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) ) such that \lim\limits _ {| \alpha | \rightarrow \infty } a _ \alpha = 0 ( | \alpha | = \sum \alpha _ {i} ). The norm on T _ {n} = T _ {n} ( K) is given by \| \sum a _ \alpha z ^ \alpha \| = \max | a _ \alpha | . The ring \{ {f \in T _ {n} } : {\| f \| \leq 1 } \} is denoted by T _ {n} ^ {o} , and T _ {n} ^ {oo} = \{ {f \in T _ {n} } : {\| f \| < 1 } \} is an ideal of T _ {n} ^ {o} . Then \widetilde{T} _ {n} = T _ {n} ^ {o} / T _ {n} ^ {oo} is easily seen to be the ring of polynomials k[ z _ {1} \dots z _ {n} ] .
The K - algebra T _ {n} ( K) is called the free Tate algebra. An affinoid algebra, or Tate algebra, A over K is a finite extension of some T _ {n} ( K) ( i.e. there is a homomorphism of K - algebras T _ {n} \rightarrow A which makes A into a finitely-generated T _ {n} - module). The space of all maximal ideals, \mathop{\rm Spm} ( A) of a Tate algebra A is called an affinoid space.
A rigid analytic space over K is obtained by glueing affinoid spaces. Every algebraic variety over K 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 K .
The theory of formal schemes over R ( the valuation ring of K ) is close to that of rigid analytic spaces. This can be seen as follows.
Fix an element \pi \in R with 0 < | \pi | < 1 . The completion of R _ {n} = R[ z _ {1} \dots z _ {n} ] with respect to the topology given by the ideals \{ {\pi ^ {m} R _ {n} } : {m> 0 } \} is the ring of strict power series R\langle z _ {1} \dots z _ {n} \rangle over R . Now R\langle z _ {1} \dots z _ {n} \rangle = T _ {n} ^ {o} , and T _ {n} ( K) is the localization of R\langle z _ {1} \dots z _ {n} \rangle with respect to \pi . So one can view \mathop{\rm Spm} ( T _ {n} ( K)) as the "general fibre" of the formal scheme \mathop{\rm Spf} ( R\langle z _ {1} \dots z _ {n} \rangle) over R . More generally, any formal scheme X over R gives rise to a rigid analytic space over K , the "general fibre" of X . Non-isomorphic formal schemes over R can have the same associated rigid analytic space over K . Further, any reasonable rigid analytic space over K is associated to some formal scheme over R .
Affinoid spaces and affinoid algebras have many properties in common with affine spaces and affine rings over K . Some of the most important are: Weierstrass preparation and division holds for T _ {n} ( K) ( cf. also Weierstrass theorem); affinoid algebras are Noetherian rings, and even excellent rings if the field K is perfect; for any maximal ideal M of an affinoid algebra A the quotient field R/M is a finite extension of K ; many finiteness theorems; any coherent sheaf S on an affinoid space \mathop{\rm Spm} ( A) is associated to a finitely-generated A - module M= H ^ {0} ( S) ( further: H ^ {i} ( S)= 0 for i \neq 0 ).
Another interpretation of T _ {n} ( K) is: T _ {n} ( K) consists of all "holomorphic functions" on the polydisc \{ {( z _ {1} \dots z _ {n} ) \in K } : {\textrm{ all } | z _ {i} | \leq 1 } \} . This interpretation is useful for finding the holomorphic functions on more complicated spaces like Drinfel'd's symmetric spaces \Omega ^ {(} n) . Let K be a local field with algebraic closure \overline{K}\; . Then
\Omega ^ {(} n) =
= \ \{ {( x _ {0} \dots x _ {n} ) \in P _ {\overline{K}\; } ^ {n} } : {\sum \lambda _ {i} x _ {i} \neq 0 \textrm{ for all } ( \lambda _ {0} \dots \lambda _ {n} ) \in P ^ {n} ( K) } \}
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=48949