TY - JOUR
T1 - Buchsbaum-Rim multiplicities as Hilbert-Samuel multiplicities
AU - Jean Chan, C. Y.
AU - Liu, Jung Chen
AU - Ulrich, Bernd
N1 - Funding Information:
* Corresponding author. Current address: Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. E-mail addresses: [email protected] (C.-Y. Jean Chan), [email protected] (J.-C. Liu), [email protected] (B. Ulrich). 1 Supported in part by the NSF.
PY - 2008/6/1
Y1 - 2008/6/1
N2 - 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].
AB - 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].
KW - Buchsbaum-Rim multiplicity
KW - Hilbert-Samuel multiplicity
KW - Linkage
KW - Reduction of ideals and modules
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U2 - 10.1016/j.jalgebra.2007.12.025
DO - 10.1016/j.jalgebra.2007.12.025
M3 - Article
AN - SCOPUS:42649131695
SN - 0021-8693
VL - 319
SP - 4413
EP - 4425
JO - Journal of Algebra
JF - Journal of Algebra
IS - 11
ER -