Equimultiple coefficient modules

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_𝔭) = e_i((B_k )_𝔭) for 1 ≤ i ≤ k and every minimal prime 𝔭 of I_r (B). The number e_i(B_𝔭) is the ith Buchsbaum-Rim coeffcient of B_𝔭 in F_𝔭. 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.

Mathematics Subject Classification (2010):

• Primary: 13H15

Key words:

• Coeffcient ideals
• integral closure
• reduction
• multiplicity