Some applications of André-Quillen homology to classes of arithmetic rings

We compute the first André-Quillen homology modules for the simple over-rings of integrally closed domains and study an ideal theoretic condition arising from the vanishing of H1. André-Quillen (co)homology is known to be a powerful tool in characterizing various classes of rings or morphisms between noetherian rings. Regular and complete intersection local rings, regular, (formally) smooth or complete intersection morphisms can be characterized with the help of this theory (see André (1974) and Brezuleanu et al. (1993)). Classes of arithmetical integral domains, such as Prüfer domains, have also been characterized in this way (see Planas-Vilanova (1996)). Let A be an integrally closed domain with quotient field K, 0 6= a, b ∈ A and B = A[a/b]. In section 1, we compute H0(A,B,B) and H1(A,B,B), and we describe H2(A,B,B) (Theorems 1.2 and 1.12). In particular, we show that H1(A,B,B) = 0 if and only if a A ∩ bA = (aA ∩ bA). In section 2, we investigate this condition in its own. We say that D is a ⋆-domain if aD ∩ bD = (aD ∩ bD) for every a, b ∈ D. The locally GCD domains are typical examples of ⋆-domains. In Proposition 2.3 we characterize the ⋆-pseudo-valuation domains. In Corollary 2.7 we show that a two-generated domain (e.g. a quadratic extension of Z) is a ⋆-domain if and only if it is Dedekind. Finally, in Proposition 2.9, we prove that the local class group of a Krull ⋆-domain has no element of order two. Throughout this paper all rings are commutative. For any undefined notation or terminology, the reader is refered to André (1974) and Gilmer (1972).