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.

Più precisamente, 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.

Contenuti 1 Enunciato del teorema 2 Discussione 3 Alternative statement 4 Guarda anche 5 References Statement of the theorem Let {displaystyle mathbf {X} =x_{1},punti ,X_{n}} denote a collection of n indeterminates, {stile di visualizzazione k[[mathbf {X} ]]} the ring of formal power series with indeterminates {displaystyle mathbf {X} } over a field k, e {displaystyle mathbf {y} =y_{1},punti ,si_{n}} a different set of indeterminates. Permettere {stile di visualizzazione f(mathbf {X} ,mathbf {y} )=0} be a system of polynomial equations in {stile di visualizzazione k[mathbf {X} ,mathbf {y} ]} , and c a positive integer. Then given a formal power series solution {stile di visualizzazione {cappello {mathbf {y} }}(mathbf {X} )in k[[mathbf {X} ]]} , there is an algebraic solution {displaystyle mathbf {y} (mathbf {X} )} consisting of algebraic functions (più precisamente, algebraic power series) tale che {stile di visualizzazione {cappello {mathbf {y} }}(mathbf {X} )equiv mathbf {y} (mathbf {X} ){in un modo {(}}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. Guarda anche: Artin's criterion.

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

Permettere {stile di visualizzazione R} be a field or an excellent discrete valuation ring, permettere {stile di visualizzazione A} be the henselization of an {stile di visualizzazione R} -algebra of finite type at a prime ideal, let m be a proper ideal of {stile di visualizzazione A} , permettere {stile di visualizzazione {cappello {UN}}} be the m-adic completion of {stile di visualizzazione A} , e lascia {displaystyle Fcolon (UN{testo{-algebras}})a ({testo{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 {stile di visualizzazione {sopra {xi }}in F({cappello {UN}})} , c'è un {displaystyle xi in F(UN)} tale che {stile di visualizzazione {sopra {xi }}equiv xi {in un modo {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, SIG 0268188 Artin, Michael (1971). Algebraic Spaces. Monografie matematiche di Yale. vol. 3. Nuovo paradiso, CT–London: Yale University Press. SIG 0407012. Raynaud, Michele (1971), "Travaux récents de M. Artin", Séminaire Nicolas Bourbaki, 11 (363): 279–295, SIG 3077132 Categorie: Moduli theoryCommutative algebraTheorems about algebras