# Artin approximation theorem

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{{Short description|1969 result in deformation theory}}
In [mathematics](/source/mathematics), the '''Artin approximation theorem''' is a fundamental result of {{harvs|last=Artin|first=Michael|txt|authorlink=Michael Artin|year=1969}} in [deformation theory](/source/deformation_theory) which implies that [formal power series](/source/formal_power_series) with coefficients in a [field](/source/field_(mathematics)) ''k'' are well-approximated by the [algebraic function](/source/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](/source/indeterminate_(variable))s, <math>k[\mathbf{x}](/source/%5Cmathbf%7Bx%7D)</math>  the [ring](/source/ring_(mathematics)) of formal [power series](/source/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 equation](/source/polynomial_equation)s in <math>k[\mathbf{x}, \mathbf{y}]</math>, and ''c'' a positive [integer](/source/integer). Then given a formal power series solution <math>\hat{\mathbf{y}}(\mathbf{x}) \in k[\mathbf{x}](/source/%5Cmathbf%7Bx%7D)</math>, there is an algebraic solution <math>\mathbf{y}(\mathbf{x})</math> consisting of [algebraic function](/source/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 space](/source/formal_moduli_space)s of deformations as [scheme](/source/scheme_(mathematics))s. See also: [Artin's criterion](/source/Artin's_criterion).

==Alternative statement==
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 [henselization](/source/Henselian_ring) 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](/source/Ring_with_the_approximation_property)
*[Popescu's theorem](/source/Popescu's_theorem)
*[Artin's criterion](/source/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](/source/Publications_Math%C3%A9matiques_de_l'IH%C3%89S) | volume=36 | issue=36 | pages=23–58| doi=10.1007/BF02684596 }}
*{{cite book|last=Artin|first= Michael|title=Algebraic Spaces|publisher= [Yale University Press](/source/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](/source/S%C3%A9minaire_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:Theorems about algebras

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