On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple ideals.
type: Item
instance of: scholarly article
title: On the mixed multiplicity and the multiplicity of blow-up rings of equimultiple ideals.
zbMATH Open document ID: 1034.13012
DOI: 10.1016/S0022-4049(03)00033-1
published in: Journal of Pure and Applied Algebra
publication date: 2003-11-17T00:00:00Z
review text: Let \(A\) be a standard graded algebra over a field and \(M\) the maximal graded ideal of \(A\). The main results of this paper (theorems 1.1 and 1.2) give formulae for the multiplicity of the Rees algebra \(A[It]\) and for mixed multiplicities of the pair \((M, I)\), where \(I\) is an arbitrary homogeneous equimultiple ideal in \(A\). However, the formulae of Theorem 1.1, Theorem 1.3 as well as Theorem 2.11, Theorem 3.1, Theorem 3.7 and their consequences are not valid in the sense that their proofs were based on a false statement, namely on the existence of weak-(FC)-sequences of homogeneous elements in \(I\) (introduced by the author). The following observation shows that such a sequence does not always exist. By the definition and lemma 2.8 of the paper, an arbitrary homogeneous weak-(FC)-sequence \(x_1,\dots,x_h\) (with respect to \((M,I)\)) satisfies the following conditions: (1) \(0:x_1 \subseteq \bigcup_{n \geq 1}0:(M I)^n\), (2) \(\deg x_1\) is the initial degree of \(I\). \noindent One can easily construct a homogeneous parameter ideal \(I\) which does not contain any homogeneous element \(x_1\) satisfying both (1) and (2), e.g. \(A = k[x,y,z] = k[X,Y,Z]/(X^2,XY)\) and \(I = (y,z^2)\). The paper also uses another false result on the maximal length of weak-(FC)-sequences in \(I\) [proved in an earlier paper; see \textit{Duong Quôc Viêt}, Proc. Am. Math. Soc. 131, 45--53 (2003; Zbl 1027.13001)]. This length is not ht(\(I)\) (as stated) but the minimal number of generators of \(I\). Since the main results of this paper hold in the case \(I\) is generated by a filter-regular sequence of non-decreasing degrees [see \textit{N. V. Trung}, J. Math. Kyoto Univ. 33, 665--683 (1993; Zbl 0816.13020) and \textit{N. D. Hoang}, Beitr. Algebra Geom. 42, 463--473 (2001; Zbl 1037.13009)], it remains to be seen whether they can be proved in general. Warning: The author said in the introduction that these results cover a similar result of Huneke-Sally on the multiplicity of R…
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13A30
zbMATH DE Number: 2005431
zbMATH Keywords: mixed multiplicity
zbMATH Keywords: Rees algebra
zbMATH Keywords: equimultiple ideal
author: Duong Quoc Viet
reviewed by: Trung Ngo Viet
MaRDI profile type: Publication
cites work: Q3247189
cites work: On the multiplicity of blow-up rings of ideals generated by \(d\)-sequences
cites work: Birational extensions in dimension two and integrally closed ideals
cites work: Q5829838
cites work: Generalizations of Reductions and Mixed Multiplicities
cites work: Q4177688
cites work: Q4047653
cites work: Filter-regular sequences and multiplicity of blow-up rings of ideals of the principal class
cites work: Positivity of mixed multiplicities
cites work: Rees Algebras and Mixed Multiplicities
cites work: Mixed multiplicities of arbitrary ideals in local rings
full work available at URL: s0022-4049(03)00033-1
OpenAlex ID: W2063994571
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description: scientific article; zbMATH DE number 2005431