# 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.

How to Cite This Entry:
Tate algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_algebra&oldid=48949
This article was adapted from an original article by M. van der Put (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article