Artin approximation theorem


In mathematics, the Artin approximation theorem is a fundamental result of 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 ; and an algebraic version of this theorem in 1969.

Statement of the theorem

Let denote a collection of n indeterminates, the ring of formal power series with indeterminates over a field k, and a different set of indeterminates. Let
be a system of polynomial equations in, and c a positive integer. Then given a formal power series solution, there is an algebraic solution consisting of algebraic functions such that

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.
Let be a field or an excellent discrete valuation ring, let be the henselization of an -algebra of finite type at a prime ideal, let m be a proper ideal of, let be the m-adic completion of, and let
be a functor sending filtered colimits to filtered colimits. Then for any integer c and any, there is a such that