Mixed multiplicity and converse of Rees' theorem for modules

• Published in: Journal of algebra 2025-02, Vol.664, p.484-510
• Main Authors: Ferrari, M.D., Jorge-Perez, V.H., Merighe, L.C.
• Format: Article
• Language: English
• Subjects: (Mixed)Buchsbaum-Rim multiplicities; Joint reduction; Modules; Reduction
• ISSN: 0021-8693
• DOI: 10.1016/j.jalgebra.2024.10.048

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 1.3. Specifically, we demonstrate the following result: Let (R,m) be a d-dimensional formally equidimensional Noetherian local ring and E1,…,Ek be finitely generated R-submodules of a free R-module F of positive rank p, with xi∈Ei for i=1,…,k. Consider S, the symmetric algebra of F, and IEi, the ideal generated by the homogeneous component of degree 1 in the Rees algebra [R(Ei)]1. Assuming that (x1,…,xk)S and IEi have the same height k and the same radical, if the Buchsbaum-Rim multiplicity of (x1,…,xk) and the mixed Buchsbaum-Rim multiplicity of the family E1,…,Ek are equal, i.e., eBR((x1,…,xk)p;Rp)=eBR(E1p,…,Ekp,Rp) for all prime ideals p minimal over ((x1,…,xk):RF), then (x1,…,xk) is a joint reduction of (E1,…,Ek). In addition to proving this theorem, we establish some properties that relate joint reduction and mixed Buchsbaum-Rim multiplicities.