Artin approximation theorem

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.

Contents 1 Statement of the theorem 2 Discussion 3 Alternative statement 4 See also 5 References 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 Fcolon (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. 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 Categories: Moduli theoryCommutative algebraTheorems about algebras