Artin Approximation via the Model Theory of Cohen-macaulay Rings

We show the existence of a first order theory Cmd,e whose Noetherian models are precisely the local Cohen-Macaulay rings of dimension d and multiplicity e. The completion of a model of Cmd,e is again a model and is moreover Noetherian. If R is an equicharacteristic local Gorenstein ring of dimension d and multiplicity e with algebraically closed residue field and if the Artin Approximation Property holds for R, then R is an existentially closed model in the subclass of all Noetherian models of Cmd,e. In case R is moreover excellent, Spivakovski proved that the weaker Henselian assumption implies the Artin Approximation. This suggests an alternative, model theoretic strategy for proving Artin Approximation under the additional assumptions that R is Gorenstein, equicharacteristic and has algebraically closed residue field. 1. Artin Approximation 1.0.1. Artin Approximation. Artin Approximation (see 1.1 for a definition) has proven to be a powerful and versatile tool in algebraic geometry and commutative algebra. Its connections with model theory have been realised by several people, but seemingly in a rather ad hoc way. Our present paper wants to provide a (as natural as possible) framework in which Artin Approximation can be studied by model theoretic tools. The key observation to link Artin Approximation with model theory is the following: a Noetherian local ring R admits the Artin Approximation property, if and only if, it is existentially closed, as a ring, in its completion R̂. (With the completion of a local ring we always mean its completion with respect to the topology given by the maximal ideal). Hence it would be highly desirable to find a first order theory T with the following two properties, where (R,m) is a Noetherian local ring. (I) If R is a model of the theory T , then so is its completion. (II) If R, moreover, admits Artin Approximation, then it is an existentially closed model of T . The theory of rings trivially verifies (I), but in general, complete (sub)theories will not. Hence (II) asks for narrowing down the models for it to become true without violating (I). The requirement (II) can never hold as it stands, as an existentially closed local ring must have an algebraically closed residue field. In Date: 24.01.99. To name few: van den Dries shows in Chapter 12 of [14] how the application of Artin Approximation by Hochster for the existence of Big Macaulay modules, can be viewed in a model theoretic way; Robinson proposes an approach to rigid analytic quantifier elimination through Artin Approximation; the author has used Artin Approximation in the study of etale complexity in [13]. Yet another instance is explained in Historic Note 1.4 below.