Artin approximation theorem: Difference between revisions

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{{Short description|1969 result in deformation theory}}
In [[mathematics]], the '''Artin approximation theorem''' is a fundamental result of [[{{harvs|last=Artin|first=Michael|txt|authorlink=Michael Artin]]|year=1969}} in [[deformation theory]] which implies that [[formal power series]] with coefficients in a [[field (mathematics)|field]] ''k'' are well-approximated by the [[algebraic function]]s on ''k''.

More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case <math>k = \Complex</math>); and an algebraic version of this theorem in 1969.

==Statement of the theorem==

Let <math>\mathbf{x} = x_1, \dots, x_n</math> denote a collection of ''n'' [[indeterminate (variable)|indeterminate]]s, <math>k[[\mathbf{x}]]</math> the [[ring (mathematics)|ring]] of formal [[power series]] with indeterminates <math>\mathbf{x}</math> over a field ''k'', and <math>\mathbf{y} = y_1, \dots, y_n</math> a different set of indeterminates. Let

Let

:<math>f(\mathbf{x}, \mathbf{y}) = 0</math>

:'''x''' = ''x''₁, …, ''x''_''n''

be a system of [[polynomial equation]]s in ''<math>k''['''\mathbf{x'''}, '''\mathbf{y'''}]</math>, and ''c'' a positive [[integer]]. Then given a formal power series solution '''ŷ'''<math>\hat{\mathbf{y}}('''\mathbf{x'''}) ∈\in ''k'''''<nowiki>[[\mathbf{x}]]</nowikimath>''', there is an algebraic solution '''<math>\mathbf{y'''}('''\mathbf{x'''})</math> consisting of [[algebraic function]]s (more precisely, algebraic power series) such that

denote a collection of ''n'' [[indeterminate (variable)|indeterminate]]s,

:<math>\hat{\mathbf{y}}(\mathbf{x}) \equiv \mathbf{y}(\mathbf{x}) \bmod (\mathbf{x})^c.</math>

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

==Discussion==

: '''y''' = ''y''₁, …, ''y''_''m''

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 moduli space]]s of deformations as [[scheme (mathematics)|scheme]]s. See also: [[Artin's criterion]].

==Alternative statement==

a different set of indeterminates. Let

The following alternative statement is given in Theorem 1.12 of {{harvs|last=Artin|first=Michael|txt|authorlink=Michael Artin|year=1969}}.

Let <math>R</math> be a field or an excellent discrete valuation ring, let <math>A</math> be the [[Henselian ring|henselization]] at a prime ideal of an <math>R</math>-algebra of finite type, let ''m'' be a proper ideal of <math>A</math>, let <math>\hat{A}</math> be the ''m''-adic completion of <math>A</math>, and let

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

:<math>F\colon (A\text{-algebras}) \to (\text{sets}),</math>

be a system of [[polynomial equation]]s in ''k''['''x''', '''y'''], and ''c'' a positive [[integer]]. Then given a formal power series solution '''ŷ'''('''x''') ∈ ''k'''''<nowiki>[[x]]</nowiki>''' there is an algebraic solution '''y'''('''x''') consisting of [[algebraic function]]s such that

be a functor sending filtered colimits to filtered colimits (Artin calls such a functor locally of finite presentation). Then for any integer ''c'' and any <math> \overline{\xi} \in F(\hat{A})</math>, there is a <math> \xi \in F(A)</math> such that

:'''ŷ'''('''x''') ≡ '''y'''('''x''') mod ('''x''')^''c''.

:<math>\overline{\xi} \equiv \xi \bmod m^c</math>.

==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 moduli space]]s of deformations as [[scheme (mathematics)|scheme]]s.

== See also ==

*[[Ring with the approximation property]]

*[[Popescu's theorem]]

*[[Artin's criterion]]

==References==

*{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebraic approximation of structures over complete local rings | url=http://www.numdam.org/item?id=PMIHES_1969__36__23_0 | mr=0268188 | year=1969 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=36 | issue=36 | pages=23–58| doi=10.1007/BF02684596 }}

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

*{{cite book|last=Artin|first= Michael|title=Algebraic Spaces|publisher= [[Yale University Press]]|series=Yale Mathematical Monographs|volume= 3|___location=New Haven, CT–London|year= 1971|mr=0407012}}

*{{citation|last=Raynaud|first= Michel|author-link=Michel Raynaud|title=Travaux récents de M. Artin| journal=[[Séminaire Nicolas Bourbaki]]|volume= 11 |year=1971|issue=363|pages= 279–295| url=http://www.numdam.org/book-part/SB_1968-1969__11__279_0/|mr=3077132}}

[[Category:Moduli theory]]

[[Category:Commutative algebra]]

[[Category:MathematicalTheorems theoremsabout algebras]]