Generalized Buchsbaum-Rim Multiplicities and a Theorem of Rees

Given M, a submodule of Nwhich is in turn a submodule of a free R-module, where Ris a Noetherian local ring which is equidimensional and universally catenary, the theorem of Rees asserts that the integral closures of Mand Nagree provided their multiplicities agree. We extend this result to modules of non-finite colength by introducing a chain of submodules H i (M)which give an increasingly good approximation to the integral closure of M, and a sequence of numbers based on the multiplicity of a pair of modules. Some applications to problems in equisingularity are given.