Artin approximation theorem

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 $k=ℂ$); and an algebraic version of this theorem in 1969.

Statement of the theorem

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

$f(𝐱,𝐲)=0$

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

$\widehat{𝐲}(𝐱)\equiv 𝐲(𝐱)mod(𝐱{)}^c.$

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

$F:(A\text{-algebras})\to (\text{sets}),$

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 $\bar{\xi }\in F(\hat{A})$, there is a $\xi \in F(A)$ such that

$\bar{\xi }\equiv \xi {modm}^c$.

See also

• Ring with the approximation property
• Popescu's theorem
• Artin's criterion