A formula for the associated Buchsbaum-Rim multiplicities of a direct sum of cyclic modules II

TL;DR

This paper derives a formula for the second to last positive Buchsbaum-Rim multiplicity of a direct sum of cyclic modules, generalizing previous results and connecting it to classical multiplicities.

Contribution

It provides a new explicit formula for the second to last Buchsbaum-Rim multiplicity in terms of known multiplicities, extending prior work by Kirby and Rees.

Findings

Derived a formula for the second to last Buchsbaum-Rim multiplicity.

Connected Buchsbaum-Rim multiplicities with Hilbert-Samuel multiplicities.

Generalized previous results to broader classes of modules.

Abstract

The associated Buchsbaum-Rim multiplicities of a module are a descending sequence of non-negative integers. These invariants of a module are a generalization of the classical Hilbert-Samuel multiplicity of an ideal. In this article, we compute the associated Buchsbaum-Rim multiplicity of a direct sum of cyclic modules and give a formula for the second to last positive Buchsbaum-Rim multiplicity in terms of the ordinary Buchsbaum-Rim and Hilbert-Samuel multiplicities. This is a natural generalization of a formula given by Kirby and Rees.