arXiv subjects

Nguyen Xuan Linh

Publications and source records attributed to Nguyen Xuan Linh.

3 recordsLinked to original sources

arXiv2026

Sequential 1-Cohen-Macaulayness for direct sums of modules

Let (R,m) be a Noetherian local ring and M1,...,Mn finitely generated R-modules. Set M is the direct sum of Mi. The main purpose of this paper is to extend the results on the sequential Cohen-Macaulayness of the direct sum of modules (Taniguchi et al, 2018), the sequential generalized Cohen-Macaulayness of the direct sum of modules (Cuong and Nhan, 2003) We first describe the largest submodule of M of dimension less than dimM by using the largest submodule so f its component modules. Then we give a necessary and sufficient condition for M being 1-Cohen-Macaulay. The purpose of this paper is to characterize the sequential 1-Cohen-Macaulayness of the direct sum M. We show that M is sequentially 1-Cohen-Macaulay if and only if Mi is sequentially 1-Cohen-Macaulay for all i <=n. We provide an example to clarify the results. We employ inductive methods as well as the dimension filtration of a finitely generated module.

math.AC↗

arXiv2026

On the pertubations of E-depth, Ef-depth and sequentially Cohen-Macaulay locus

Let (R,m) be a Noetherian local ring, I an ideal of R and M a finitely generated R-module. In this paper, we define and study E-depth and Ef-depth of M in I. We prove that E-depth (resp. Ef-depth under mild conditions) of M in I is the common length of all maximal sequential sequences (resp. maximal sequential f-sequences) of M in I. We show that E-depth and Ef-depth do not decrease under small pertubations. We describe the non sequentially Cohen-Macaulay locus nSCM(M) of M in terms of support and non Cohen-Macaulay locus of deficiency modules of M. Finally, we show that the dimension of non sequentially Cohen-Macaulay locus with respect to a sequential f-sequence does not increase under small pertubations.

math.AC↗

arXiv2025

On sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules

We introduce the notions of sequential sequence and sequential f-sequence in order to characterize sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules. Let R be a Noetherian local ring and M a finitely generated R-module. We show that M is sequentially Cohen-Macaulay (resp. sequentially generalized Cohen-Macaulay) if and only if there exists a system of parameters of M that is an M- sequential sequence (resp. each generalized regular sequence s.o.p of M is an M-sequential f-sequence) and R/AnnR(M) is a quotient of a Cohen-Macaulay local ring. As an application, we give new characterizations of Cohen-Macaulay modules and generalized Cohen-Macaulay modules.

math.AC↗