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

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

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

:<math>\hat{\mathbf{y}}(\mathbf{x})  \equiv \mathbf{y}(\mathbf{x}) \bmod (\mathbf{x})^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 spaces of deformations as [[Scheme (mathematics)|scheme]]s. See also: [[Artin's criterion]].

==Alternative statement==
The following alternative statement is given in Theorem 1.12 of {{harvs|last=Artin|first=Michael|txt|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 

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

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 

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

== See also ==
*Ring with the approximation property
*[[Popescu's theorem]]
*[[Artin's criterion]]

==References==
*{{Citation | last1=Artin | first1=Michael |  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]] | issue=36 | pages=23–58}}
*{{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|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:Commutative algebra]]

{{Sourceattribution|Artin approximation theorem}}