The minimal number of generators of an invertible ideal

Abstract

All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let Rhe a ring with total quotient ring K. A fractional ideal of -R is invertible if II-1 = R; equivalently, is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I-1 = {R:I) = {x ∈ K\xl ⊆ R}. Moreover, a projective i?-module of rank 1 is isomorphic to an invertible ideal. (We use the term "ideal" in the sense of an integral ideal.) We denote the minimal number of generators of an ideal I oi R by vR{I).li R is a Dedekind domain, equivalently, a domain in which each nonzero ideal is invertible, then vR{I) ≤ 2 for each nonzero ideal ; moreover, is strongly 2- generated, in the sense that one of the generators can be an arbitrary nonzero element of I. A Dedekind domain is characterized as an integrally closed Noetherian domain of Krull dimension 1. It turns out that of these three properties, Krull dimension 1 always implies that an invertible ideal is 2- generated, as was shown by Sally and Vasconcelos in [SV74]. R. Heitmann generalized this fact to arbitrary finite Krull dimension: an invertible ideal of an n-dimensional domain R is strongly (n + l)-generated (see Section 3). Moreover, this result is sharp, in the sense that for each n ≤ 1 there exists an n-dimensional domain R, even Priifer, with an invertible ideal requiring n + l generators: see the examples in Sections 4, 5 and 6. The general problem of determining the minimal number of generators for an invertible ideal of a domain was first studied by Gilmer and Heinzer in [GH70]. Among other fundamental results, they provide sufficient conditions for an invertible ideal to have the property that it can be generated by two elements. The question whether a finitely generated ideal of a Priifer domain can be always be generated by 2 elements was first raised by Gilmer around 1964 [Swa84]. Recall that a Priifer domain is an integral domain in which each nonzero finitely generated ideal is invertible. In a 1979 paper Schiilting gave an example of a Priifer domain with an invertible ideal that cannot be generated by 2 elements. Schiilting's result was generalized by Swan and Kucharz. We discuss in Sections 4 and 5 the different approaches used by these authors. In Section 4 we give a new, more direct proof of Schiilting's example, and in Section 5, we elaborate on some steps in the proof of Kucharz's theorem on holomorphy rings of finitely generated ideals. In doing so we highlight his method for constructing in a formally real function field of degree n > 0 over a real closed field an invertible ideal in the holomorphy ring that cannot be generated by n elements.