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

{\displaystyle k=\mathbb {C} }

); and an algebraic version of this theorem in 1969.

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).