Artin approximation theorem

{{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 ''k'' are well-approximated by the algebraic functions 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'' indeterminates, <math>k\mathbf{x}</math> the 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

:<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. 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 functions (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 schemes. See also: 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 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 | 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 }} *{{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:Theorems about algebras