Artin approximation theorem

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In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin (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]\displaystyle{ k = \Complex }[/math]); and an algebraic version of this theorem in 1969.

Statement of the theorem

Let [math]\displaystyle{ \mathbf{x} = x_1, \dots, x_n }[/math] denote a collection of n indeterminates, [math]\displaystyle{ k\mathbf{x} }[/math] the ring of formal power series with indeterminates [math]\displaystyle{ \mathbf{x} }[/math] over a field k, and [math]\displaystyle{ \mathbf{y} = y_1, \dots, y_n }[/math] a different set of indeterminates. Let

[math]\displaystyle{ f(\mathbf{x}, \mathbf{y}) = 0 }[/math]

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

[math]\displaystyle{ \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 Michael Artin (1969).

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

[math]\displaystyle{ 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]\displaystyle{ \overline{\xi} \in F(\hat{A}) }[/math], there is a [math]\displaystyle{ \xi \in F(A) }[/math] such that

[math]\displaystyle{ \overline{\xi} \equiv \xi \bmod m^c }[/math].

See also

References