arXiv subjects

Le Thanh Nhan

Publications and source records attributed to Le Thanh Nhan.

9 recordsLinked to original sources

arXiv2026

Buchsbaum modules and parameter ideals under flat extensions

Let $(R,\frak m)$ and $(S, \frak n)$ be commutative Noetherian local rings, let $φ: R\to S$ be a flat local homomorphism. In this paper, we first characterize the ascent and descent of Buchsbaum modules and generalized Cohen-Macaulay modules via the flat extension $φ$. Then, we provide relationships between the parameter ideals of a finitely generated $R$-module $M$ and those of $M\otimes_RS$ under the assumption that $\frak{n} = \frak{m} S.$

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

Annihilator of local cohomology modules under localization and completion

Let $(R, \frak m)$ be a Noetherian local ring. This paper deals with the annihilator of Artinian local cohomology modules $H^i_{\frak m}(M)$ in the relation with the structure of the base ring $R$, for non negative integers $i$ and finitely generated $R$-modules $M$. Firstly, the catenarity and the unmixedness of local rings are characterized via the compatibility of annihilator of top local cohomology modules under localization and completion, respectively. Secondly, some necessary and sufficient conditions for a local ring being a quotient of a Cohen-Macaulay local ring are given in term of the annihilator of all local cohomology modules under localization and completion.

math.AC↗

arXiv2025

Ascent and descent of Artinian module structures under flat base changes

Let $φ: R\rightarrow S$ be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between $R$ and $S$ are investigated. For an Artinian $R$-module $A$, the structure of $A\otimes_RS$ is described. As an application, the Artinianess of certain local cohomology modules is clarified.

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↗

arXiv2020

Reducibility index and sum-reducibility index

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, Northcott introduced the reducibility index of $M$, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ in $M$. On the other hand, for an Artinian $R$-module $A$, Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of $A$ does not depend on the choice of the representation. This number is called the sum-reducibility index of $A$. In the former part of this paper, we compute the reducibility index of $S\otimes_R M$, where $R\to S$ is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of $R$ are studied. For the latter part of this paper, we clarify the relation among the reducibility index of $M$, that of the completion of $M$, and the sum-reducibility index of the Matlis dual of $M$.

math.AC↗

arXiv2014

Attached primes of local cohomology modules under localization and completion

Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. Following I. G. Macdonald \cite{Mac}, the set of all attached primes of the Artinian local cohomology module $H^i_{\m}(M)$ is denoted by $\Att_R(H^i_{\m}(M))$. In \cite[Theorem 3.7]{Sh}, R. Y. Sharp proved that if $R$ is a quotient of a Gorenstein local ring then the shifted localization principle always holds true, i.e. $$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Att_{R_{\p}}\big(H^{i-\dim (R/\p)}_{\p R_{\p}}(M_{\p})\big)=\big\{\q R_{\p}\mid \q\in\Att_RH^i_{\m}(M), \q\subseteq \p\big\} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$ for any local cohomology modules $H^i_{\m}(M)$ and any $\p\in\Spec (R).$ In this paper, we improve Sharp's result as follows: the shifted localization principle always holds true if and only if $R$ is universally catenary and all its formal fibers are Cohen-Macaulay, if and only if $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \displaystyle \Att_{\R}(H^i_{\m}(M))=\bigcup_{\p\in\Att_R(H^i_{\m}(M))}\Ass_{\R}(\R/\p\R)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ holds true for any finitely generated $R$-module $M$ and any integer $i\geq 0.$ This also improves the main result of the paper \cite{CN}.

math.AC↗

arXiv2010

On pseudo supports and non Cohen-Macaulay locus of finitely generated modules

Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module with $\dim M=d.$ Let $i\geq 0$ be an integer. Following M. Brodmann and R. Y. Sharp \cite{BS1}, the $i$-th pseudo support of $M$ is the set of all prime ideals $\p$ of $R$ such that $ H^{i-\dim (R/\p)}_{\p R_{\p}}(M_{\p})\neq 0.$ In this paper, we study the pseudo supports and the non Cohen-Macaulay locus of $M$ in connections with the catenarity of the ring $R/\Ann_RM$, the Serre conditions on $M$, and the unmixedness of the local rings $R/\p$ for certain prime ideals $\p$ in $\Supp_R (M)$.

math.AC↗

arXiv2007

Top local cohomology and the catenaricity of the unmixed support of a finitely generated module

Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R-$module with $\dim M=d.$ This paper is concerned with the following property for the top local cohomology module $H^d_\m(M)$: $$\Ann (0:_{H^d_\m(M)}\p)=\p\ \text{for all prime ideals} \p\supseteq\Ann H^d_\m(M).$$ In this paper we will show that this property is equivalent to the catenaricity of the unmixed support $\Usupp M$ of $M$ which is defined by $\Usupp M=\Supp M/U_M(0)$, where $U_M(0)$ is the largest submodule of $M$ of dimension less than $d.$ Some characterizations of this property in terms of system of parameters as well as the relation between the unmixed supports of $M$ and of the $\m$-adic completion $\hat M$ are given.

math.AC↗