Artin approximation theorem

In mathematics, the Artin approximation theorem is a fundamental result of {{harvs|last=Artin|first=Michael|txt|authorlink=Michael Artin|year=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 $k = \Complex$ ); and an algebraic version of this theorem in 1969.

Statement of the theorem

Let $\mathbf{x} = x_1, \dots, x_n$ denote a collection of n indeterminates, $k \mathbf{x}$ the ring of formal power series with indeterminates $\mathbf{x}$ over a field k, and $\mathbf{y} = y_1, \dots, y_n$ a different set of indeterminates. Let

: $f(\mathbf{x}, \mathbf{y}) = 0$

be a system of polynomial equations in $k[\mathbf{x}, \mathbf{y}]$ , and c a positive integer. Then given a formal power series solution $\hat{\mathbf{y}}(\mathbf{x}) \in k \mathbf{x}$ , there is an algebraic solution $\mathbf{y}(\mathbf{x})$ consisting of algebraic functions (more precisely, algebraic power series) such that

: $\hat{\mathbf{y}}(\mathbf{x}) \equiv \mathbf{y}(\mathbf{x}) \bmod (\mathbf{x})^c.$

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 {{harvs|last=Artin|first=Michael|txt|authorlink=Michael Artin|year=1969}}.

Let $R$ be a field or an excellent discrete valuation ring, let $A$ be the henselization at a prime ideal of an $R$ -algebra of finite type, let m be a proper ideal of $A$ , let $\hat{A}$ be the m-adic completion of $A$ , and let

: $F\colon (A\text{-algebras}) \to (\text{sets}),$

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 $\overline{\xi} \in F(\hat{A})$ , there is a $\xi \in F(A)$ such that

: $\overline{\xi} \equiv \xi \bmod m^c$ .

References

{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebraic approximation of structures over complete local rings | url=http://www.numdam.org/item?id=PMIHES_1969__36__23_0 | mr=0268188 | year=1969 | journal=Publications Mathématiques de l'IHÉS | volume=36 | issue=36 | pages=23–58| doi=10.1007/BF02684596 }}
{{cite book|last=Artin|first= Michael|title=Algebraic Spaces|publisher= Yale University Press|series=Yale Mathematical Monographs|volume= 3|location=New Haven, CT–London|year= 1971|mr=0407012}}
{{citation|last=Raynaud|first= Michel|author-link=Michel Raynaud|title=Travaux récents de M. Artin| journal=Séminaire Nicolas Bourbaki|volume= 11 |year=1971|issue=363|pages= 279–295| url=http://www.numdam.org/book-part/SB_1968-1969__11__279_0/|mr=3077132}}

Category:Moduli theory

Category:Commutative algebra

Category:Theorems about algebras