Joint reductions and mixed Buchsbaum-Rim multiplicities of modules and a joint-reduction-number-zero theorem

TL;DR

This paper introduces new concepts of joint reductions and mixed Buchsbaum-Rim multiplicities for modules, providing proofs of a key theorem and linking these multiplicities to Euler-Poincaré characteristics and intersection theory.

Contribution

It presents novel definitions of joint reductions and mixed Buchsbaum-Rim multiplicities, along with two proofs of a joint-reduction-zero theorem and connections to intersection theory.

Findings

New definitions of joint reductions and mixed Buchsbaum-Rim multiplicity.

Two different proofs of the joint-reduction-zero theorem.

Relation of multiplicities to Euler-Poincaré characteristic and intersection theory.

Abstract

We offer new definitions of joint reductions and mixed Buchsbaum-Rim multiplicity for certain collections of modules over a Noetherian local ring and illustrate their application to give two different proofs of a joint-reduction-number-zero theorem for integrally closed modules over two-dimensional regular local rings. We also relate the mixed Buchsbaum-Rim multiplicity of modules to the Euler-Poincar\'{e} characteristic of a natural Koszul complex and relate it to the mixed Buchsbaum-Rim multiplicity of ideals by generalising a lemma from intersection theory.