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

Statement of the theorem

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Let   denote a collection of n indeterminates,   the ring of formal power series with indeterminates   over a field k, and   a different set of indeterminates. Let

 

be a system of polynomial equations in  , and c a positive integer. Then given a formal power series solution  , there is an algebraic solution   consisting of algebraic functions (more precisely, algebraic power series) such that

 

Discussion

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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

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The following alternative statement is given in Theorem 1.12 of Michael Artin (1969).

Let   be a field or an excellent discrete valuation ring, let   be the henselization at a prime ideal of an  -algebra of finite type, let m be a proper ideal of  , let   be the m-adic completion of  , and let

 

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  , there is a   such that

 .

See also

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References

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  • 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