Mixed multiplicity and Converse of Rees' theorem for modules

TL;DR

This paper proves a converse to Rees' mixed multiplicity theorem for modules, extending classical results and establishing conditions under which certain modules have joint reductions based on multiplicity equality.

Contribution

It extends Rees' mixed multiplicity theorem to modules, providing new criteria for joint reductions using Buchsbaum-Rim multiplicities.

Findings

Proves the converse of Rees' mixed multiplicity theorem for modules.

Establishes conditions linking joint reduction and multiplicity equality.

Provides properties relating joint reduction and mixed Buchsbaum-Rim multiplicities.

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.