Curves and coherent Prüfer rings

Usual definitions of Dedekind domain are not well suited for an algorithmic treatment. Indeed, the notion of Noetherian rings is subtle from a constructive point of view, and to be able to get prime ideals involve strong hypotheses. For instance, if k is a field, even given explicitely, there is in general no method to factorize polynomials in k[X]. The work [2] analyses the notion of Dedekind domain from a constructive point of view. A first good constructive approximation of the notion of Dedekind domain is the notion of coherent Prüfer ring1. We recall the required definitions. Classically, a ring R is arithmetical iff any localisation Rp at any prime p of R is a valuation ring, i.e. such that the divisibility relation is linear. A ring R is arithmetical iff its lattice of ideal is distributive iff for any pair of elements x, y we can find u, v, w such that xv = yu and x(1− u) = yw. Yet another equivalent definition, which can be seen as a formal version of the classical definition is that for any pair of elements x, y we can find a covering D(w1), . . . , D(wn) of the Zariski spectrum of R such that x divides y or y divides x in each localisation Rwi . We say that a ring is a Prüfer ring iff all its ideal are flat iff it is arithmetical and reduced (if x2 = 0 then x = 0). One can then show that a Prüfer ring is coherent (i.e. any finitely generated ideal is finitely presented) iff it is a pp-ring (i.e. the annihilator of any element is generated by an idempotent)2. In particular any domain which is arithmetical is a coherent Prüfer ring. However to assume the ring to be integral is too strong constructively since we cannot decide irreducibility in general. The goal of this paper is to show, in constructive mathematics, that if k is a discrete field and f an arbitrary polynomial in k[x, y] then the localisation Rf ′ y is always a coherent Prüfer ring3, where R denotes the ring k[x, y] quotiented by f . (Computationally, this means in particular that we have to solve the following problem: given g, h two elements of k[x, y] to find u0 = g, v0 = h, u1, v1, . . . , un, vn in k[x, y] such that vig = uih modulo f for i = 0, . . . , n and D(f ′ y) is covered by D(u0), D(v0), . . . , D(un), D(vn) in the Zariski spectrum of R.) An important corollary is that R is a coherent Prüfer ring whenever 1 = 〈f, f ′ x, f ′ y〉. We first give a simple argument in the case where k is algebraically closed and f is irreducible. As a preliminary to the general case, we present after a generalisation of the notion of Hasse-