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 = x₁, …, 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 = y₁, …, y_m

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 ŷ(x) ∈ k[[x]] there is an algebraic solution y(x) consisting of algebraic functions (more precisely, algebraic power series) such that

ŷ(x) ≡ 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 of an R-algebra of finite type at a prime ideal, let m be a proper ideal of A, let ${\hat {A}}$ be the m-adic completion of A, and let

F: (A-algebras) → (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 ${\overline {\xi }}\in F({\hat {A}})$ there is a $\xi \in F(A)$ such that

{\overline {\xi }}

≡

\xi

mod m^c.

References

Artin, Michael (1969), "Algebraic approximation of structures over complete local rings", Publications Mathématiques de l'IHÉS (36): 23–58, MR 0268188
Artin, Michael. Algebraic Spaces. Yale University Press, 1971.

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