Hilbert coefficients and partial Euler-Poincaré characteristics of Koszul complexes of d-sequences
type: Item
instance of: scholarly article
title: Hilbert coefficients and partial Euler-Poincaré characteristics of Koszul complexes of d-sequences
zbMATH Open document ID: 1334.13019
author: Phạm Hong Nam
author: Trung Cuòng Đoàn
published in: Journal of Algebra
publication date: 2015-09-04T00:00:00Z
review text: In the paper under review the notion of ``almost \(p\)-standard system of parameters'' is defined. In [\textit{N. T. Cuong} and \textit{D. T. Cuong}, J. Algebra Appl. 6, No. 2, 207--231 (2007; Zbl 1183.13027)] it is shown that such a system of parameter is the same as \(dd\)-sequence which in turn is a generalization of \(d\)-sequence. A sequence \(x_1,\dots,x_r \in \mathfrak{m}\) is a \(dd\)-sequence on finitely generated \(R\)-module \(M\) if \(x_1^{n_1}, \dots, x_i^{n_i}\) is a \(d\)-sequence on the module \(M/(x_{i+1}^{n_{i+1}}, \dots, x_r^{n_r})M\) for \(i=1,\dots,r\) and any \(n_1, \dots, n_r>0.\) There is a characterization for rings which are homomorphic image of a Cohen-Macaulay module in terms of \(dd\)-sequences or equivalently in terms of \(p\)-standard system of parameters'', see (Theorem 1.2, 1.3, \textit{N. T. Cuong} and \textit{D. T. Cuong} [``Local cohomology annihilators and Macaulayfication'', \url{arXiv:1305.5609}]). In the other point of view almost \(p\)-standard sequence can be defined as a generalization of \(p\)-standard sequence of type 0 due to \textit{T. Kawasaki} (Definition 2.2, [Trans. Am. Math. Soc. 360, No. 5, 2709--2739 (2008; Zbl 1137.13009)]). Using almost \(p\)-standard system of parameters \(x_1, \dots, x_d\) of a module \(M\), certain sub-quotients \(U^{i}_M\) of \(M\) (\(1\leq i \leq d+1\)) are defined which are independent of the choice of the system of parameter (see Proposition 3.2) and in several interesting and technical theorems the authors establish relations between multiplicities of these modules and the following invariants; {\parindent=6mm \begin{itemize} \item[{\(\bullet\)}] Length of \(M/(x_1^{n_1}, \dots, x_d^{n_d})M\) for any \(n_1, \dots, n_d>0\). (Theorem 3.7) \item [{\(\bullet\)}] \(\chi_{k}(x_1, \dots, x_d;M)\) partial Euler-Poicaré characteristic. (Theorem 4.5) \item [{\(\bullet\)}] Hilbert coefficients \(e_{d-i}(x_1, \dots, x_d;M)\). (Theorem 4.7) \end{itemize}}
reviewed by: M. Mostafazadehfard
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 6478588
zbMATH Keywords: partial Euler-Poincaré characteristics
zbMATH Keywords: Hilbert coefficient
zbMATH Keywords: d-sequence
zbMATH Keywords: dd-sequence
zbMATH Keywords: almost p-standard system of parameters
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.06.024
OpenAlex ID: W2213399098
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DOI: 10.1016/J.JALGEBRA.2015.06.024
description: scientific article; zbMATH DE number 6478588