ec 1 99 3 Complete ideals defined by sign conditions and the real spectrum of a two - dimensional local ring

0. Introduction. Let (A,m) be a regular local ring and let α and β be points of the real spectrum of A centered at m . α and β may be viewed as total orderings on quotients of A . Associated with α and β , there is the so-called “separating ideal”, 〈α, β〉 ⊆ A , which is generated by all a ∈ A such that a is non-negative with respect to α and −a is non-negative with respect to β . 〈α, β〉 is a valuation ideal for the valuation canonically associated with α (or β ), and hence is complete. It is known that a thorough understanding of 〈α, β〉 would contribute greatly to a solution of the long-standing PierceBirkhoff conjecture (see [M]) but up to now no good techniques for working with it have been known. In this paper, we investigate 〈α, β〉 by applying quadratic transforms to (A,m) and using Zariski’s theory of complete ideals in two-dimensional regular local rings (see [Z]) to analyze how 〈α, β〉 is affected. Suppose that (A,m) is a quadratic transform of A . Under natural hypotheses, α and β induce points α and β in the real spectrum of A which are centered at m . Our main result, Theorem 4.7, is a formula which relates 〈α, β〉 ⊆ A with the ideal transform 〈α, β〉 of 〈α, β〉 in A . It says that if A is twodimensional and has real closed residue field, and if 〈α, β〉 is not the maximal ideal, then 〈α, β〉 = 〈α, β〉 . Applications of this result are presented in detail elsewhere (see [MR] and [MS]). The applications show that the transformation formula provides an essentially complete understanding of separating ideals in two dimensional regular algebras over real closed fields. Here is a summary of the contents. In Section 1, we review notation for valuations and for the real spectrum, and then make some observations on valuations induced by points of the real spectrum. In Section 2, we prove (Proposition 2.2) that separating ideals are simple if a certain technical condition is satisfied. In section 3, we examine the behavior of the real spectrum under quadratic transformation, and we prove some general facts about the effect of quadratic transformations on separating ideals. We also give an example which indicates that in dimensions 3 and higher the behavior of separating ideals under quadratic transformations will be very difficult to analyze. In section 4, we consider 2dimensional regular local rings, and we prove the transformation formula mentioned above. With Zariski’s theory and the results of sections 1 through 3 at hand, the hardest part of the proof of the transformation formula is Theorem 4.4. A sampling of some of the results which will be presented in [MR] and [MS] is given in the last section. The present paper had a rather lengthly gestation. Alvis and Madden worked together on the type of ideals studied here in 1989, but were unaware of Zariski’s work on complete ideals at that time. Alvis wrote several computer programs which searched for generators for separating ideals. Without the wealth of examples found this way, connections to quadratic transforms would not have been noticed later on. Madden learned about complete ideals in two-dimensional rings from Johnston in 1990-91, and together they conjectured a version of Theorem 4.7 in early 1991. The proof of 4.7 was completed