Hilbert coefficients and partial Euler-Poincaré characteristics of Koszul complexes of d-sequences
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}}
scientific article; zbMATH DE number 6478588
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