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Journal of the London Mathematical Society Advance Access published online on October 9, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm054
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© 2007 London Mathematical Society

Mixed Multiplicities for Arbitrary Ideals and Generalized Buchsbaum-Rim Multiplicities

R. Callejas-Bedregal and V. H. Jorge Pérez

Universidade Federal da Paraíba-DM
Cidade Universitária
58051-900 João Pessoa, PB
Brazil
roberto{at}mat.ufpb.br
Universidade de São Paulo-ICMC
Av. do Trabalhador São Carlense - Centro
Caixa Postal 668
13560-970 São Carlos, SP
Brazil

We introduce first the notion of mixed multiplicities for arbitrary ideals in a local d-dimensional Noetherian ring (A,m) which, in some sense, generalizes the concept of mixed multiplicities for m-primary ideals. We also generalize Teissier's product formula for a set of arbitrary ideals and extend the notion of the Buchsbaum-Rim multiplicity (BR-multiplicity) of a submodule of a free module to the case where the submodule no longer has finite colength. For a submodule M of Ap, we introduce a sequence of multiplicities ekBR (M), k = 0, . . . , d + p – 1 which in the case of an ideal (p = 1) coincides with the multiplicity sequence c0(I,A), . . . , cd(I,A) defined for an arbitrary ideal I of A by Achilles and Manaresi. In the case where M has finite colength in Ap and is totally decomposable, we prove that our BR-multiplicity sequence essentially falls into the standard BR-multiplicity of M.

2000 Mathematics Subject Classification 14C17 (primary), 13A30 (secondary).

The first author was partially supported by PADCT/CT-INFRA/CNPq/MCT grant 620120/04-5 and the second author was partially supported by CAPES grant 3131/04-1 and PADCT/CT-INFRA/CNPq/MCT grant620120/04-5.

Received December 7, 2005; revised August 24, 2006;
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