### Abstract

We study the Buchsbaum-Rim multiplicity br (M) of a finitely generated module M over a regular local ring R of dimension 2 with maximal ideal m. The module M under consideration is of finite colength in a free R-module F. Write F / M ≅ I / J, where J ⊂ I are m-primary ideals of R. We first investigate the colength ℓ (R / a) of any m-primary ideal a and its Hilbert-Samuel multiplicity e (a) using linkage theory. As an application, we establish several multiplicity formulas that express the Buchsbaum-Rim multiplicity of the module M in terms of the Hilbert-Samuel multiplicities of ideals related to I, J and a minimal reduction of M. The motivation comes from work by E. Jones, who applied graphical computations of the Hilbert-Samuel multiplicity to the Buchsbaum-Rim multiplicity [E. Jones, Computations of Buchsbaum-Rim multiplicities, J. Pure Appl. Algebra 162 (2001) 37-52].

Original language | English |
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Pages (from-to) | 4413-4425 |

Number of pages | 13 |

Journal | Journal of Algebra |

Volume | 319 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2008 Jun 1 |

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### Keywords

- Buchsbaum-Rim multiplicity
- Hilbert-Samuel multiplicity
- Linkage
- Reduction of ideals and modules

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*319*(11), 4413-4425. https://doi.org/10.1016/j.jalgebra.2007.12.025