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.

Etwas präziser, 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.

Inhalt 1 Aussage des Theorems 2 Diskussion 3 Alternative statement 4 Siehe auch 5 References Statement of the theorem Let {Anzeigestil mathbf {x} =x_{1},Punkte ,x_{n}} denote a collection of n indeterminates, {Anzeigestil k[[mathbf {x} ]]} the ring of formal power series with indeterminates {Anzeigestil mathbf {x} } over a field k, und {Anzeigestil mathbf {j} =y_{1},Punkte ,y_{n}} a different set of indeterminates. Lassen {Anzeigestil f(mathbf {x} ,mathbf {j} )=0} be a system of polynomial equations in {Anzeigestil k[mathbf {x} ,mathbf {j} ]} , and c a positive integer. Then given a formal power series solution {Anzeigestil {Hut {mathbf {j} }}(mathbf {x} )in k[[mathbf {x} ]]} , there is an algebraic solution {Anzeigestil mathbf {j} (mathbf {x} )} consisting of algebraic functions (etwas präziser, algebraic power series) so dass {Anzeigestil {Hut {mathbf {j} }}(mathbf {x} )Äquiv. mathbf {j} (mathbf {x} ){in gewisser Weise {(}}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. Siehe auch: Artin's criterion.

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

Lassen {Anzeigestil R} be a field or an excellent discrete valuation ring, Lassen {Anzeigestil A} be the henselization of an {Anzeigestil R} -algebra of finite type at a prime ideal, let m be a proper ideal of {Anzeigestil A} , Lassen {Anzeigestil {Hut {EIN}}} be the m-adic completion of {Anzeigestil A} , und lass {displaystyle Fcolon (EIN{Text{-algebras}})zu ({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 {Anzeigestil {überstreichen {xi }}in F({Hut {EIN}})} , da ist ein {displaystyle xi in F(EIN)} so dass {Anzeigestil {überstreichen {xi }}equiv xi {in gewisser Weise {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, HERR 0268188 Artin, Michael (1971). Algebraic Spaces. Yale Mathematische Monographien. Vol. 3. Neuer Hafen, CT–London: Yale University Press. HERR 0407012. Raynaud, Michel (1971), "Travaux récents de M. Artin", Séminaire Nicolas Bourbaki, 11 (363): 279–295, HERR 3077132 Kategorien: Moduli theoryCommutative algebraTheorems about algebras