Tate algebra

Let $K$ 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 -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