Mixed multiplicities and the minimal number of generator of modules

Abstract

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.