Ultraproducts and approximation in local rings I

Abstract

In this paper we give new and quite elementary proofs of the strong approximation theorems of M. Greenberg (Theorem 2.1), M. Artin (Theorem 3.2) and D. Popescu (Theorem 3.3). Our proofs use the ultraproduct construction. In w ! we give a brief outline of this construction and derive the properties we shall use. Our method of proof gives a quite general way of deriving a strong approximation theorem from the corresponding approximation theorem. (By an "approximation theorem" we mean a theorem of the type of Theorem 3.1 and by a "strong approximation theorem" we mean a theorem of the type of Theorem 3.2 or 3.3.) For example if we have an approximation theorem with some of the Y's constrained to depend only on some of the X's then we can immediately derive the strong approximation theorem with the same constraints. In w 4 we give a pair of such theorems with the constraints that some of the Y's depend only on X 1. In w 5 we present some counterexamples: Let k be the algebraic closure of the field with p elements. There is a system of polynomials f ( X t, X2, Yt . . . . . Yv) over k such that the system f (X1, X2, Yt .... ,Y v)=0; Y1, Y6ek[[_X2~; Y2, Y3, Y6ek[[XI~ has a solution mod (X1, X2) J, for every j e N but has no solution in k[[_Xz, X2~. We also give a similar counterexample over the rationals. In w 6 we discuss effectivity. In Ultraproducts and Approximation in Local Rings II we shall extend the methods of this paper to give proofs of some further strong approximation theorems, e.g. the Theorem of Pfister and Popescu [17] as well as some new results. The usefulness of the ultraproduct construction (or equivalently an appropriate form of the G~Sdel Completeness and Compactness Theorems) in analyzing algebraic questions about valued fields is due to Ax and Kochen [3] (cf. also