Artin approximation theorem

In mathematics, the Artin approximation theorem is a fundamental result of 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 $\mathbf = x_1, \dots, x_n$ denote a collection of ''n'' indeterminates, k \mathbf the ring of formal power series with indeterminates $\mathbf$ over a field ''k'', and $\mathbf = y_1, \dots, y_n$ a different set of indeterminates. Let

:$f(\mathbf, \mathbf) = 0$

be a system of polynomial equations in k mathbf, \mathbf/math>, and ''c'' a positive integer. Then given a formal power series solution \hat(\mathbf) \in k \mathbf, there is an algebraic solution $\mathbf(\mathbf)$ consisting of algebraic functions (more precisely, algebraic power series) such that

:$\hat(\mathbf) \equiv \mathbf(\mathbf) \bmod (\mathbf)^c.$

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 .

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

:$F\colon (A\text) \to (\text),$

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

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

See also

* Ring with the approximation property
*Popescu's theorem
* Artin's criterion

References

*
*
*{{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

Moduli theory
Commutative algebra
Theorems about algebras