View source for Tate algebra

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922401.png" /> be a field which is complete with respect to an [[ultrametric]] valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922402.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922403.png" />). The valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922404.png" /> has a unique maximal ideal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922405.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922406.png" /> is called the residue field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922407.png" />.

Examples of such fields are the local fields, i.e. finite extensions of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922408.png" />-adic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t0922409.png" />, or the field of Laurent series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224011.png" /> with coefficients in the finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224012.png" /> (cf. also [[Local field|Local field]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224013.png" /> denote indeterminates. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224014.png" /> denotes the algebra of all power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224016.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224017.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224018.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224019.png" />). The norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224020.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224021.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224022.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224024.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224025.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224026.png" /> is easily seen to be the ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224027.png" />.

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224028.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224029.png" /> is called the free Tate algebra. An affinoid algebra, or Tate algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224030.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224031.png" /> is a finite extension of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224032.png" /> (i.e. there is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224033.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224034.png" /> which makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224035.png" /> into a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224036.png" />-module). The space of all maximal ideals, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224037.png" /> of a Tate algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224038.png" /> is called an affinoid space.

A [[rigid analytic space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224039.png" /> is obtained by glueing affinoid spaces. Every algebraic variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224040.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224041.png" />.

The theory of formal schemes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224042.png" /> (the valuation ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224043.png" />) is close to that of rigid analytic spaces. This can be seen as follows.

Fix an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224044.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224045.png" />. The completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224046.png" /> with respect to the topology given by the ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224047.png" /> is the ring of strict power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224048.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224049.png" />. Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224051.png" /> is the localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224052.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224053.png" />. So one can view <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224054.png" /> as the  "general fibre"  of the formal scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224055.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224056.png" />. More generally, any formal scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224057.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224058.png" /> gives rise to a rigid analytic space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224059.png" />, the  "general fibre"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224060.png" />. Non-isomorphic formal schemes over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224061.png" /> can have the same associated rigid analytic space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224062.png" />. Further, any reasonable rigid analytic space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224063.png" /> is associated to some formal scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224064.png" />.

Affinoid spaces and affinoid algebras have many properties in common with affine spaces and affine rings over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224065.png" />. Some of the most important are: Weierstrass preparation and division holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224066.png" /> (cf. also [[Weierstrass theorem|Weierstrass theorem]]); affinoid algebras are Noetherian rings, and even excellent rings if the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224067.png" /> is perfect; for any maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224068.png" /> of an affinoid algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224069.png" /> the quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224070.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224071.png" />; many [[Finiteness theorems|finiteness theorems]]; any coherent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224072.png" /> on an affinoid space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224073.png" /> is associated to a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224074.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224075.png" /> (further: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224076.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224077.png" />).

Another interpretation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224078.png" /> is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224079.png" /> consists of all  "holomorphic functions"  on the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224080.png" />. This interpretation is useful for finding the holomorphic functions on more complicated spaces like Drinfel'd's symmetric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224081.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224082.png" /> be a local field with algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224083.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224084.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224085.png" /></td> </tr></table>

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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bosch,   U. Güntzer,   R. Remmert,   "Non-Archimedean analysis" , Springer  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.G. Drinfel'd,   "Coverings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224086.png" />-adic symmetric regions"  ''Funct. Anal. Appl.'' , '''10''' :  2  (1976)  pp. 107–115  ''Funkts. Anal. Prilozhen.'' , '''10''' :  2  pp. 29–41</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Faltings,   "Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten" , ''Lect. notes in math.'' , '''1111''' , Springer  (1984)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Fresnel,   M. van der Put,   "Géométrie analytique rigide et applications" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Gerritzen,   M. van der Put,   "Schottky groups and Mumford curves" , ''Lect. notes in math.'' , '''817''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. Mumford,   "An analytic construction of degenerating curves over complete local fields"  ''Compos. Math.'' , '''24'''  (1972)  pp. 129–174</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Mumford,   "An analytic construction of degenerating abelian varieties over complete rings"  ''Compos. Math.'' , '''24'''  (1972)  pp. 239–272</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  D. Mumford,   "An algebraic surface with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224087.png" /> ample, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092240/t09224089.png" />"  ''Amer. J. Math.'' , '''101'''  (1979)  pp. 233–244</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Raynaud,   "Variétés abéliennes en géométrie rigide" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''2''' , Gauthier-Villars  (1971)  pp. 473–477</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J. Tate,   "Rigid analytic spaces"  ''Invent. Math.'' , '''12'''  (1971)  pp. 257–289</TD></TR></table>