Mixed multiplicity and Converse of Rees' theorem for modules

Abstract: 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.