Space of Valuations

This paper illustrates the general program of applying point-free topology to constructive mathematics, and especially abstract commutative algebra [5, 4, 6, 7, 10, 21], on the notion of valuations. Historically, it can be argued that it is, well before Stone [22], one of the first example of point-free, and algebraic, representation of spaces [3]. The notion of valuation was indeed introduced implicitely by Dedekind and Weber [13] to give a rigourous and (almost) algebraic presentation of some ideas of Riemann. In particular, the starting point is a finite algebraic extension K of C(X), intuitively thought of as the field of meromorphic functions over a Riemann surfaces, and their idea can be described as defining the points of a Riemann surface as the (discrete) valuations of the field K1. As explained in the survey [5], we think that ideas from locale theory [22, 21] and general facts from logic, especially the completeness theorem of first-order logic, and its relativised version for coherent logic, are powerful tools for analysing the logical structure of abstract algebra. The present work may be seen as a partial realisation Hilbert’s program in algebra, the general idea being to see the notion of “ideal” elements of Hilbert as a point of a space, and the point-free description of this space as realising the “elimination of ideal elements” required by Hilbert. It is then remarkable that the classical arguments, which seem to rely on non constructive existence of abstract objects, can be interpreted almost as they are and provide then constructive proofs that could have been accepted by Kronecker [8, 5]. Contrary to what one may think however, we have not found many examples where the use of classical mathematics shorten the proofs. Our experience has instead been that, besides using weaker logical tools, the constructive proof is often clearer and shorter than the corresponding classical arguments which rely on points (a typical example is provided by [9]). It should be pointed out however that the present work may suggest also examples where Boolean logic allows to shorten considerably the statements and proofs: this appears in Lemmas 6.2 and 6.4. This paper is organised as follow. We first recall basic notions related to distributive lattices as a pointfree representation of formal spaces. We present then the Zariski lattice associated to a commutative ring, which is a pointfree presentation of the Zariski spectrum, a construction due to Joyal [21]. By analogy, we introduce the main object of the paper, which is the space of valuations associated to any field. We show that many of the results of [20] can be naturally expressed and proved in this point free framework. We use then these results to give a point free description of the notion of algebraic curve, which as we explained above, go really back to [13, 23]. In particular, we show how the cohomological description [29] of the genus of a curve, a notion which goes back to Abel [17], can also be interpreted constructively. We