Mixed multiplicities of arbitrary modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring. In this work we extend the notion of mixed multiplicities of modules, given in \cite{Kleiman-Thorup2} and \cite{Kirby-Rees1} (see also \cite{Bedregal-Perez}), to an arbitrary family $E,E_1,\mathrm{.}\mathrm{.}\mathrm{.},E_q$ of $R$-submodules of $R^p$ with $E$ of finite colength. We prove that these mixed multiplicities coincide with the Buchsbaum-Rim multiplicity of some suitable $R$-module. In particular, we recover the fundamental Rees’s mixed multiplicity theorem for modules, which was proved first by Kirby and Rees in \cite{Kirby-Rees1} and recently also proved by the authors in \cite{Bedregal-Perez}. Our work is based on, and extend to this new context, the results on mixed multiplicities of ideals obtained by Vi^et in \cite{Viet8} and Manh and Vi^et in \cite{Manh-Viet}. We also extend to this new setting some of the main results of Trung in \cite{Trung} and Trung and Verma in \cite{Trung-Verma1}. As in \cite{Kleiman-Thorup2}, \cite{Kirby-Rees1} and \cite{Bedregal-Perez}, we actually work in the more general context of standard graded $R$-algebras.