Artin approximation theorem: Difference between revisions

Revision as of 00:07, 7 August 2005

Revision as of 11:25, 28 October 2005

Line 2:

==Statement of the theorem==

Let '''x''' = ''x''₁, &hellip;, ''x''_''n'' denote a collection of ''n'' [[indeterminate (variable)|indeterminate]]s, ''k''['''x'''] the [[ring (mathematics)|ring]] of formal power series with indeterminants '''x''' over a field ''k'', '''y''' = ''y''₁, &hellip;, ''y''_''m'' a different set of indeterminants, ''f''('''x''', '''y''') = 0 a system of polynomial equations over ''k''['''x''', '''y'''], and ''c'' a positive [[integer]]. Then given a formal power series solution '''ŷ'''('''x''') &isin; ''k''[['''x''']] there is an algebraic solution '''y'''('''x''') consisting of algebraic functions such that '''ŷ'''('''x''') &equiv; '''y'''('''x''') mod ('''x''')<sup>''c''<sup>.

Let

:'''x''' = ''x''₁, &hellip;, ''x''_''n'' denote a collection of ''n'' [[indeterminate (variable)|indeterminate]]s,

''k''['''x'''] the [[ring (mathematics)|ring]] of formal power series with indeterminates '''x''' over a field ''k'',

: '''y''' = ''y''₁, &hellip;, ''y''_''m''

a different set of indeterminates. Let

:''f''('''x''', '''y''') = 0

be a system of [[polynomial equation]]s over ''k''['''x''', '''y'''], and ''c'' a positive [[integer]]. Then given a formal power series solution '''ŷ'''('''x''') &isin; ''k''[['''x''']] there is an algebraic solution '''y'''('''x''') consisting of algebraic functions such that

:'''ŷ'''('''x''') &equiv; '''y'''('''x''') mod ('''x''')<sup>''c''<sup>.

==Discussion==

Given any desired positive integer ''c'', this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by ''c''. This leads to theorems that deduce the existence of certain formal [[formal moduli space]]s of deformations as [[scheme (mathematics)|scheme]]s.

==Reference==

Artin, Michael. ''Algebraic Spaces''. Yale University Press, 1971.

*Artin, Michael. ''Algebraic Spaces''. Yale University Press, 1971.

[[Category:Algebraic geometry]]

[[Category:AlgebraicModuli geometrytheory]]

[[Category:Commutative algebra]]

[[Category:Theorems]]