Mixed multiplicities and the minimal number of generator of modules

Let (R, m) be a d-dimensional Noetherian local ring. In this work we prove that the mixed Buchsbaum-Rim multiplicity for a finite family of R-submodules of R(p) of finite colength coincides with the Buchsbaum-Rim multiplicity of the module generated by a suitable superficial sequence, that is, we generalize for modules the well-known Risler-Teissier theorem. As a consequence, we give a new proof of a generalization for modules of the fundamental Rees` mixed Multiplicity theorem, which was first proved by Kirby and Rees in (1994, [8]). We use the above result to give an upper bound for the minimal number of generators of a finite colength R-submodule of R(p) in terms of mixed multiplicities for modules, which generalize a similar bound obtained by Cruz and Verma in (2000, [5]) for m-primary ideals. (C) 2009 Elsevier B.V. All rights reserved.

FAPESP-Brazil[08/53703-0]

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Procad/Capes[190/2007]

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

CNPq-Brazil[308915/2006-2]