Integrally closed modules over two-dimensional regular local rings

Abstract

This paper is based on work of Rees on integral closures of modules and initiates the study of integrally closed modules over two-dimensional regular local rings in analogy with the classical theory of complete ideals of Zariski. The main results can be regarded as generalizations of Zariski’s product theorem. They assert that the tensor product mod torsion of integrally closed modules is integrally closed, that the symmetric algebra mod torsion of an integrally closed module is a normal domain and that the first Fitting ideal of an integrally closed module is an integrally closed ideal. A construction of indecomposable integrally closed modules is also given. The primary technical tool is a study of the Buchsbaum-Rim multiplicity.