Artin approximation theorem

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Deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. Theorem as a fundamental result of Michael Artin

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 {\displaystyle k=\mathbb {C} } ); and an algebraic version of this theorem in 1969.

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Statement of the theorem

Let x = x 1 , … , x n {\displaystyle \mathbf {x} =x_{1},\dots ,x_{n}} denote a collection of n indeterminates, k [ [ x ] ] {\displaystyle k[[\mathbf {x} ]]} the ring of formal power series with indeterminates x {\displaystyle \mathbf {x} } over a field k, and y = y 1 , … , y n {\displaystyle \mathbf {y} =y_{1},\dots ,y_{n}} a different set of indeterminates. Let

f ( x , y ) = 0 {\displaystyle f(\mathbf {x} ,\mathbf {y} )=0}

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

y ^ ( x ) ≡ y ( x ) mod ( x ) c . {\displaystyle {\hat {\mathbf {y} }}(\mathbf {x} )\equiv \mathbf {y} (\mathbf {x} ){\bmod {(}}\mathbf {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 {\displaystyle R} be a field or an excellent discrete valuation ring, let A {\displaystyle A} be the henselization at a prime ideal of an R {\displaystyle R} -algebra of finite type, let m be a proper ideal of A {\displaystyle A} , let A ^ {\displaystyle {\hat {A}}} be the m-adic completion of A {\displaystyle A} , and let

F : ( A -algebras ) → ( sets ) , {\displaystyle F\colon (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 ξ ¯ ∈ F ( A ^ ) {\displaystyle {\overline {\xi }}\in F({\hat {A}})} , there is a ξ ∈ F ( A ) {\displaystyle \xi \in F(A)} such that

ξ ¯ ≡ ξ mod m c {\displaystyle {\overline {\xi }}\equiv \xi {\bmod {m}}^{c}} .