On the multiplicity of blow-up rings of ideals generated by \(d\)-sequences
type: Item
instance of: scholarly article
title: On the multiplicity of blow-up rings of ideals generated by \(d\)-sequences
zbMATH Open document ID: 0778.13001
DOI: 10.1016/0022-4049(92)90146-7
author: Trung Ngo Viet
author: Bernd Ulrich
author: Jürgen Herzog
published in: Journal of Pure and Applied Algebra
publication date: 1993-01-16T00:00:00Z
review text: Let \(R=R_ 0+R_ 1+R_ 2+\dots\) be a noetherian graded ring generated by \(R_ 1\) over \(R_ 0=k\), a field, and let \(I\) be an ideal generated by a homogeneous \(d\)-sequence \(x_ 1,\dots,x_ n\). Let \(S=R[It]\) be the Rees ring, \(S[t^{-1}]\) be the extended Rees ring and gr\(_ I(R)=S/IS\) be the associated graded ring. The aim of the paper is to calculate the multiplicities (with respect to their respective unique maximal graded ideal) of these ``blow-up rings''. Set \(a_ i=\deg x_ i\) and \(I_ i=(x_ 1,\dots,x_{i-1}):x_ i\), \(1\leq i\leq n\). Assume that \(a_ 1\leq a_ 2\leq\cdots\leq a_ n\). Then the multiplicities \(e(S)\) etc. are explicitly calculated in terms of \(e(R/I_ i)\)'s and the degrees \(a_ i\). In particular, if \(\dim R/I=\dim R-n\), then one gets very simple results like \(e(S)=(1+a_ 1+a_ 1a_ 2+\cdots+a_ 1a_ 2\dots a_{n-1})\cdot e(R)\), \(e(\text{gr}_ I(R))=a_ 1\dots a_{n- 1}e(R)\). The proofs are as interesting as the results. The authors use certain filtration \({\mathcal F}\) on \(S\), finer than the adic filtration with respect to the maximal graded ideal, and get the nice presentation gr\(_{\mathcal F}(S)\simeq R[T_ 1,\dots,T_ n]/(I_ 1T_ 1,\dots,I_ nT_ n)\). The technique is basically similar to that of Gröbner basis.
reviewed by: H.Matsumura
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 89585
zbMATH Keywords: Cohen-Macaulay ring
zbMATH Keywords: \(d\)-sequence
zbMATH Keywords: blow-up rings
zbMATH Keywords: graded ring
zbMATH Keywords: extended Rees ring
zbMATH Keywords: associated graded ring
zbMATH Keywords: multiplicities
zbMATH Keywords: filtration
MaRDI profile type: Publication
full work available at URL: 0022-4049(92)90146-7
OpenAlex ID: W2074449531
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description: scientific article; zbMATH DE number 89585