Mixed multiplicity and Converse of Rees' theorem for modules

In this paper, we prove the converse of Rees' mixed multiplicity theorem for modules, which extends the converse of the classical Rees' mixed multiplicity theorem for ideals given by Swanson - Theorem \ref{SwansonTheorem}. Specifically, we demonstrate the following result: Let $(R,\mathfrak{m})$ be a $d$-dimensional formally equidimensional Noetherian local ring and $E_1,\dots,E_k$ be finitely generated $R$-submodules of a free $R$-module $F$ of positive rank $p$, with $x_i\in E_i$ for $i=1,\dots,k$. Consider $S$, the symmetric algebra of $F$, and $I_{E_i}$, the ideal generated by the homogeneous component of degree 1 in the Rees algebra $[\mathscr{R}(E_i)]_1$. Assuming that $(x_1,\ldots,x_k)S$ and $I_{E_i}$ have the same height $k$ and the same radical, if the Buchsbaum-Rim multiplicity of $(x_1,\dots,x_k)$ and the mixed Buchsbaum-Rim multiplicity of the family $E_1,\dots,E_k$ are equal, i.e., ${\rm e_{BR}}((x_1,\dots,x_k)_{\mathfrak{p}};R_{\mathfrak{p}}) = {\rm e_{BR}}({E_1}_{\mathfrak{p}},\dots, {E_k}_{\mathfrak{p}},R_{\mathfrak{p}})$ for all prime ideals $\mathfrak{p}$ minimal over $((x_1,\ldots,x_k):_RF)$, then $(x_1,\ldots,x_k)$ is a joint reduction of $(E_1,\dots,E_k)$. In addition to proving this theorem, we establish several properties that relate joint reduction and mixed Buchsbaum-Rim multiplicities.