Buchsbaum-Rim multiplicities as Hilbert-Samuel multiplicities

C. Y. Jean Chan*, Jung Chen Liu, Bernd Ulrich

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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 languageEnglish
Pages (from-to)4413-4425
Number of pages13
JournalJournal of Algebra
Issue number11
Publication statusPublished - 2008 Jun 1


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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