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

Statement of the theorem

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

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

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

${\displaystyle {\hat {\mathbf {y} }}(\mathbf {x} )\equiv \mathbf {y} (\mathbf {x} ){\bmod {(}}\mathbf {x} )^{c}.}$

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

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

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

${\displaystyle {\overline {\xi }}\equiv \xi {\bmod {m}}^{c}}$.

See also

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

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. 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