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

Statement of the theorem

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

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

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

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

{\overline {\xi }}\equiv \xi {\bmod {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 (1971). Algebraic Spaces. Yale Mathematical Monographs. Vol. 3. New Haven, CT–London: Yale University Press. MR 0407012.
Raynaud, Michel (1971), "Travaux récents de M. Artin", Séminaire Nicolas Bourbaki, 11 (363): 279–295, MR 3077132