Extended Rees algebras and mixed multiplicities
type: Item
instance of: scholarly article
title: Extended Rees algebras and mixed multiplicities
zbMATH Open document ID: 0655.13027
DOI: 10.1007/BF01180686
author: Daniel Katz
author: J. K. Verma
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174078
review text: Let \((R,m)\) be a local ring of positive dimension \(d\). Let \(I\) be an \(m\)-primary ideal and \(J\) an ideal of positive height with analytic spread \(a\). It is proved that the coefficient of \(r^{d-1-i}s i\) in the polynomial \(\ell (I\) rJ \(s/I^{r+1}J s)\) for large \(r\) and \(s\) is nonzero if and only if \(i=0,1,...,a-1\). A formula is developed for the multiplicity of the ideal \((t^{-1},I,Jt)\) in the extended Rees algebra \(R[Jt,t^{-1}]\) localized at \((t^{-1},m,Jt)\). A criterion is given for the Cohen-Macaulayness of \(R[It,t^{-1}]\). Using these results, the class of parameter ideals \(I\) in \(R\) is completely determined for \(R[It,t^{-1}]\) to be Cohen-Macaulay with minimal multiplicity at its maximal homogeneous ideal \((t^{-1},m,It)\).
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 4069053
zbMATH Keywords: Bhattacharya polynomial
zbMATH Keywords: mixed multiplicities
zbMATH Keywords: local ring
zbMATH Keywords: analytic spread
zbMATH Keywords: extended Rees algebra
zbMATH Keywords: Cohen-Macaulayness
zbMATH Keywords: parameter ideals
MaRDI profile type: Publication
cites work: Local Rings of High Embedding Dimension
cites work: Q3247189
cites work: Five theorems on Macaulay rings
cites work: Birational extensions in dimension two and integrally closed ideals
cites work: Q3781197
cites work: A conjecture of Nagata on graded Cohen-Macaulay rings
cites work: Q5829838
cites work: Q3847036
cites work: Rings associated with ideals and analytic spread
cites work: Generalizations of Reductions and Mixed Multiplicities
cites work: On the associated graded ring of a local Cohen-Macaulay ring
cites work: Q4170834
cites work: Rees Algebras and Mixed Multiplicities
cites work: Certain graded algebras are always Cohen-Macaulay
cites work: On form rings which are Cohen-Macaulay
OpenAlex ID: W2036018541
description: scientific article; zbMATH DE number 4069053