Let A be a finitely generated commutative algebra over a field K with a presentation A = K〈X1, . . . ,Xn | R〉, where R is a set of monomial relations in the generators X1, . . . ,Xn. Necessary and sufficient conditions are found for A to be an integrally closed domain provided that the presentation involves at most two relations. The class group of such algebras A is calculated. Examples are gi...

Proof. We will calculate the integral closure of A inside K. We begin with a more general discussion. Suppose C is an irreducible, affine curve, that is, C is some irreducible affine variety of dimension 1. Let A be the ring of regular functions on C, so that A is a noetherian domain of Krull dimension 1. Localization preserves the noetherian condition, as well as the domain condition. Thus, th...

Let A be a noetherian ring which is locally Macaulay. For each integer i ^ O , groups d(A) and Wi(A) are denned, each sequence of groups generalizing to higher dimensions the usual class group of an integrally closed noetherian domain. d(A) is called the ith class group of A, and Wi(A) is called the ith homologuai class group of A. The main purpose of this note is to show that both sequences of...