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.

Mais precisamente, Artin proved two such theorems: one, dentro 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.

Conteúdo 1 Declaração do teorema 2 Discussão 3 Alternative statement 4 Veja também 5 References Statement of the theorem Let {estilo de exibição mathbf {x} =x_{1},pontos ,x_{n}} denote a collection of n indeterminates, {estilo de exibição k[[mathbf {x} ]]} the ring of formal power series with indeterminates {estilo de exibição mathbf {x} } over a field k, e {estilo de exibição mathbf {y} =y_{1},pontos ,s_{n}} a different set of indeterminates. Deixar {estilo de exibição f(mathbf {x} ,mathbf {y} )=0} be a system of polynomial equations in {estilo de exibição k[mathbf {x} ,mathbf {y} ]} , and c a positive integer. Then given a formal power series solution {estilo de exibição {chapéu {mathbf {y} }}(mathbf {x} )in k[[mathbf {x} ]]} , there is an algebraic solution {estilo de exibição mathbf {y} (mathbf {x} )} consisting of algebraic functions (mais precisamente, algebraic power series) de tal modo que {estilo de exibição {chapéu {mathbf {y} }}(mathbf {x} )equiv mathbf {y} (mathbf {x} ){de uma maneira {(}}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. Veja também: Artin's criterion.

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

Deixar {estilo de exibição R} be a field or an excellent discrete valuation ring, deixar {estilo de exibição A} be the henselization of an {estilo de exibição R} -algebra of finite type at a prime ideal, let m be a proper ideal of {estilo de exibição A} , deixar {estilo de exibição {chapéu {UMA}}} be the m-adic completion of {estilo de exibição A} , e deixar {displaystyle Fcolon (UMA{texto{-algebras}})para ({texto{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 {estilo de exibição {overline {XI }}in F({chapéu {UMA}})} , existe um {displaystyle xi in F(UMA)} de tal modo que {estilo de exibição {overline {XI }}equiv xi {de uma maneira {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. Monografias Matemáticas de Yale. Volume. 3. New Haven, CT–London: Imprensa da Universidade de Yale. MR 0407012. Raynaud, Michel (1971), "Travaux récents de M. Artin", Séminaire Nicolas Bourbaki, 11 (363): 279-295, MR 3077132 Categorias: Moduli theoryCommutative algebraTheorems about algebras