Space of Valuations

The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [28, 12, 33]. Formal topology presents a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. Points are then infinite ideal objects, defined as particular filters of neighborhoods, while basic open sets are thought of as primitive symbolic objects or observable facts [13]. This is a reverse of the traditional conceptual order in topology which defines opens as particular sets of points [33]. Some roots of this approach involve Brouwer’s notion of choice sequences, and an analysis of the status of infinite objects and of universal quantification over these objects in constructive mathematics [30]1. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionistic mathematics, which correspond to Hilbert’s required elimination of ideal objects2. Such a technique has been used in infinitary combinatorics, obtaining intuitionistic versions of highly non constructive arguments [4, 5, 6]. More recently, several works [7, 9, 10, 11, 16, 18, 27] following these ideas can be seen as achieving a partial realization of Hilbert’s program in the field of commutative algebra. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [17] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [3]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. In this work with Weber [17], Dedekind used his newly created theory of ideals, a theory that has played an important rôle in the development of non constructive methods in mathematics [19, 21]. It is thus also relevant to illustrate Hilbert’s notion of introduction and elimination of ideal elements in this context. Our work relies here directly on [18], which pointed out the notion of Prüfer domain as the right constructive (and first-order) approximation of Dedekind rings. We extend this work and present several characterization of Prüfer domains. We think that some of our proofs illustrate well Hilbert’s ideas of elimination of ideal elements. The points (prime ideals, valuations, . . .) constitute a powerful intuitive help, but they are used here only as suggestive means with no actual existence. We show that many of the results of [26, 35] can be naturally expressed and proved in this point-free framework, illustrating