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.

Plus précisément, Artin proved two such theorems: one, dans 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.

Contenu 1 Énoncé du théorème 2 Discussion 3 Alternative statement 4 Voir également 5 References Statement of the theorem Let {style d'affichage mathbf {X} =x_{1},des points ,X_{n}} denote a collection of n indeterminates, {style d'affichage k[[mathbf {X} ]]} the ring of formal power series with indeterminates {style d'affichage mathbf {X} } over a field k, et {style d'affichage mathbf {y} =y_{1},des points ,y_{n}} a different set of indeterminates. Laisser {style d'affichage f(mathbf {X} ,mathbf {y} )=0} be a system of polynomial equations in {style d'affichage k[mathbf {X} ,mathbf {y} ]} , and c a positive integer. Then given a formal power series solution {style d'affichage {chapeau {mathbf {y} }}(mathbf {X} )in k[[mathbf {X} ]]} , there is an algebraic solution {style d'affichage mathbf {y} (mathbf {X} )} consisting of algebraic functions (plus précisément, algebraic power series) tel que {style d'affichage {chapeau {mathbf {y} }}(mathbf {X} )mathbf équiv {y} (mathbf {X} ){dans un sens {(}}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. Voir également: Artin's criterion.

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

Laisser {style d'affichage R} be a field or an excellent discrete valuation ring, laisser {style d'affichage A} be the henselization of an {style d'affichage R} -algebra of finite type at a prime ideal, let m be a proper ideal of {style d'affichage A} , laisser {style d'affichage {chapeau {UN}}} be the m-adic completion of {style d'affichage A} , et laissez {displaystyle Fcolon (UN{texte{-algebras}})à ({texte{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 {style d'affichage {surligner {xii }}in F({chapeau {UN}})} , Il y a un {displaystyle xi in F(UN)} tel que {style d'affichage {surligner {xii }}equiv xi {dans un sens {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, M 0268188 Artin, Michael (1971). Algebraic Spaces. Monographies mathématiques de Yale. Volume. 3. Nouveau Havre, CT–London: Presse de l'Université de Yale. M 0407012. Raynaud, michel (1971), "Travaux récents de M. Artin", Séminaire Nicolas Bourbaki, 11 (363): 279–295, M 3077132 Catégories: Moduli theoryCommutative algebraTheorems about algebras