Complete intersections which are abelian extensions of a factorial domain (Q1922156)

This paper is concerned with the following question: Let \(A\) be a noetherian local factorial domain, let \(K\) be its field of fractions and let \(L\) be an abelian extension of \(K\). The integral closure of \(A\) in \(L\) is called an abelian extension of \(A\). One asks when an abelian extension \(R\) of \(A\) is a complete intersection. By suitably adapting a method of K.-i. Watanabe, the author gives a condition in terms of the so called datum, a concept slightly too technical to be defined here. The paper is divided into four sections. The first two sections are devoted to establishing the framework in the class of \(G\)-graded rings, where \(G\) is an abelian group. Here, the intended group is the Galois group of the abelian extension \(L/K\) as the abelian extension \(R\) is naturally graded by this group. In these sections the author gives the analogues of various constructions by M. Demazure, M. Tomari and K.-i. Watanabe. The main proofs are in section 3. These are notationally involved but quite natural and, perhaps unavoidable. It would seem as if the given criterion had mainly a theoretical interest rather than a powerful test. However, once one knows that \(R\) is a complete intersection then it is very nice to see the explicit defining equations as an \(A\)-algebra.