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[edit]
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
![{\displaystyle k[[\mathbf {x} ]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d65b542a1a24f34a921dac394d502d48ea3ff61)
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
![{\displaystyle k[\mathbf {x} ,\mathbf {y} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51ea2d1b2a983e591c2bb19add98f879d81aec0)
![{\displaystyle {\hat {\mathbf {y} }}(\mathbf {x} )\in k[[\mathbf {x} ]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d2f105ed93b4721e5c08ffe1fc06bb57ecdcca5)
Discussion[edit]
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[edit]
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[edit]
References[edit]
- 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