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Buchsbaum-Rim Multiplicities as Hilbert-Samuel Multiplicities

C-Y. Jean Chan·Jung-Chen Liu·Bernd Ulrich
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Abstract

Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and its Hilbert-Samuel multiplicity using linkage theory. As applications, we establish several multiplicity formula that express the Buchsbaum-Rim multiplicity of M in terms of the Hilbert-Samuel multiplicities of ideals related to a certain minimal reduction U of M. Moreover, there exists m-primary Bourbaki ideals I and J of the modules F and M respectively such that F/M is isomorphic to I/J. We also have a formula for the Buchsbaum-Rim multiplicity whenever such I and J are given. The latter part is related to a graphical interpretation of the multiplicities in the case of monomial ideals.

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C-Y. Jean Chan, Jung-Chen Liu, Bernd Ulrich. 2006-06-17. Buchsbaum-Rim Multiplicities as Hilbert-Samuel Multiplicities. https://arxiv.org/abs/math/0606413

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