| مستخلص: |
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 E 1 , ... , E k be finitely generated R -submodules of a free R -module F of positive rank p , with x i ∈ E i for i = 1 , ... , 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 [ R (E i) ] 1. Assuming that (x 1 , ... , x k) S and I E i have the same height k and the same radical, if the Buchsbaum-Rim multiplicity of (x 1 , ... , x k) and the mixed Buchsbaum-Rim multiplicity of the family E 1 , ... , E k are equal, i.e., e BR ((x 1 , ... , x k) p ; R p) = e BR ( E 1 p , ... , E k p , R p) for all prime ideals p minimal over ((x 1 , ... , x k) : R F) , then (x 1 , ... , x k) is a joint reduction of (E 1 , ... , E k). In addition to proving this theorem, we establish some properties that relate joint reduction and mixed Buchsbaum-Rim multiplicities. [ABSTRACT FROM AUTHOR] |