Le Thanh Nhan

Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules

Abstract In this paper we study the structure of two classes of modules called pseudo Cohen–Macaulay and pseudo generalized Cohen–Macaulay modules. We also give a characterization for these modules in term of the Cohen–Macaulayness and generalized Cohen–Macaulayness. Then we apply this result to prove a cohomological characterization for sequentially Cohen–Macaulay and sequentially generalized Cohen–Macaulay modules.

ON GENERALIZED REGULAR SEQUENCES AND THE FINITENESS FOR ASSOCIATED PRIMES OF LOCAL COHOMOLOGY MODULES

Abstract Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R -module. The two notions of generalized regular sequence and generalized depth are introduced as extensions of the known notions of regular sequence and depth, respectively. Some properties of generalized regular sequence and generalized depth, which are closely related to those of regular sequence and depth, are given. If x1 ,…, xr is a generalized regular sequence of M , then is a finite set. Some finiteness properties for associated primes of local cohomology modules are presented.

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 : In this article 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/UM(0), where UM(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 are given.

On the length of generalized fractions

Abstract Let M be a finitely generated module over a Noetherian local ring (R, m ) with dimM=d. Let (x1,…,xd) be a system of parameters of M and (n1,…,nd) a set of positive integers. Consider the length of generalized fraction 1/(x1n1,…,xdnd,1) as a function in n1,…,nd. Sharp and Hamieh [J. Pure Appl. Algebra 38 (1985) 323–336] asked whether this function is a polynomial for n1,…,nd large enough. In this paper, we will give counterexamples to this question. We also study conditions on the system of parameters x , in order to show that the length of the generalized fraction 1/(x1n1,…,xdnd,1) is not a polynomial for n1,…,nd large enough.

Generalized F-Modules and the Associated Primes of Local Cohomology Modules

ABSTRACT We introduce a class of modules called generalized f-modules, which contains strictly all f-modules and generalized Cohen–Macaulay modules. The properties of multiplicity, local cohomology modules, localization, completion… of these modules are presented. A result concerning the finiteness of associated primes of local cohomology modules with respect to generalized f-modules is given. Some connections to the coordinate rings of algebraic varieties and Stanley-Reisner rings are considered.

On representable linearly compact modules

For a flat R-module F, we prove that Hom R (F, -) is a functor from the category of linearly compact R-modules to itself and is exact. Moreover, Hom R (F, M) is representable when M is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization Hom R (R S , M) of a linearly compact representable R-module M with respect to a multiplicative set S in R is described.

Non-Cohen-Macaulay Locus and Non Generalized Cohen-Macaulay Locus

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. In this paper, we improve and extend some known relations among the non-Cohen–Macaulay locus nCM(M), the polynomial type p(M) and the annihilators of local cohomology modules . A characterization for the base ring R being universally catenary with all Cohen–Macaulay formal fibers is given. The nongeneralized Cohen–Macaulay locus nGCM(M) of M is described.

A Finiteness Result for Attached Primes of Certain Tor-Modules

In this paper we introduce the notion of width in dimension > s for Artinian modules and give a finiteness result for attached primes of certain Tor-modules.

Generalized Co-Cohen–Macaulay and Co-Buchsbaum Modules

We study two classes of Artinian modules called co-Buchsbaum modules and generalized co-Cohen–Macaulay modules. Some basic properties and characterizations of these modules in terms of 𝔮-weak co-sequences, co-standard sequences, multiplicity, local homology modules are presented.