Review

Equimultiple coefficient modules

Published in: Quaestiones Mathematicae
Volume 43 , issue 2 , 2020 , pages: 283–292

DOI: 10.2989/16073606.2019.1577308

Author(s): P.H. Lima Federal University of Maranhão, Brazil , V.H. Jorge Pérez Institute of Mathematics and Computer Science, ICMC, University of São Paulo, Brazil

Keywords: Primary: 13H15 , Primary: 13H15

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Abstract

Let (R, m) be a d-dimensional quasi-unmixed Noetherian local ring and B ⊆ F := R^r an equimultiple finitely generated R-module of rank r and analytic spread s. Let I_r (B) denote the 0-th fitting ideal of F/B. In this paper, we use Buchsbaum-Rim polynomial of B ⊆ F to prove the existence of a chain of modules B ⊆ B_s ⊆ ··· ⊆ B₁ ⊆

, between B and its integral closure , where B_k is the unique largest submodule of F containing B such that e_i(B_????) = e_i((B_k )_????) for 1 ≤ i ≤ k and every minimal prime ???? of I_r (B). The number e_i(B_????) is the ith Buchsbaum-Rim coeffcient of B_???? in F_????. The module B_k is called the kth equimultiple coeﬃcient module of B ⊆ F. We obtain B_s = ()^u, the unmixed part of the Ratliﬀ-Rush module . In fact, we prove that each B_k is an unmixed Ratliﬀ-Rush module.

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