A linear formula for the generalized multiplicity sequence

2012/11/14 by Thomas Boyd Dunn, Dunn, Thomas
Computer Science · Mathematics · #13H15 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1211.3192

openalex publication_date 2012/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an arbitrary ideal I in a local ring R and a finitely generated R-module M, we prove a formula expressing each generalized multiplicity sequence c_k(I,M) as a linear combination of certain local multiplicities. As a consequence, when M is formally equidimensional, we prove that if I is contained in J and c_k(I,M)=c_k(J,M) for all k, then I is a reduction of (J,M). The converse of this statement is also known to be true by a result of Ciuperca. This theorem gives a complete numerical characterization of the integral closure, generalizing a well known theorem of Rees.

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