Английская Википедия:Artin approximation theorem

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Шаблон:Short description In mathematics, the Artin approximation theorem is a fundamental result of Шаблон:Harvs 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 Шаблон:Harvs.

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

• Шаблон:Citation
• Шаблон:Cite book
• Шаблон:Citation