Nguyen Tu Cuong

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.

The I-adic completion and local homology for Artinian modules

Let I be an ideal of a commutative ring R and M an R -module. It is well known that the I -adic completion functor Λ I defined by Λ I ( M ) = lim ← t M / I t M is an additive exact covariant functor on the category of finitely generated R -modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, Λ I is neither left nor right exact on the category of all R -modules. Nevertheless, we can consider the sequence of left derived functors { L I i } of Λ I , in which L I 0 is right exact, but in general L I 0 ≠ Λ I . Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R -regular sequence, Matlis proved in [ 9 , 10 ] that where D (−) = Hom R (−; E ( R /[mfr ])) is the Matlis dual functor, and that In [ 18 , 5 ] A.-M. Simon shows that L I o (M) = M and L I i ( M ) = 0 for i > 0, provided that M is complete with respect to the I -adic topology. Later, Greenlees and May [ 3 ] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R ) by where x is a finitely generated system of I . Then they showed, under some conditions on x which are satisfied when R is Noetherian, that H I [bull ] ( M ) ≅ L I [bull ] ( M ). Recently, Tarrio, Lopez and Lipman [ 1 ] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.

On the dimension of the non-Cohen–Macaulay locus of local rings admitting dualizing complexes

In this paper we mainly consider local rings admitting dualizing complexes. It is well-known that if a Noetherian local ring A admits a dualizing complex, then the non-Cohen–Macaulay (abbreviated CM) locus of A is closed in the Zariski topology (cf. [8, 10]). If the dimension of this locus is zero and A is equidimensional, i.e. the punctured spectrum of A is locally CM and dim( A/P ) = dim ( A ) for all minimal prime ideals P ∈ Ass ( A ), then A is a generalized CM ring and its structure is well-understood (see [2, 12]). For instance, one of the characterizations of generalized CM rings is the conditions that for any parameter ideal q contained in a large power of the maximal ideal m of A , the difference between length and multiplicity is independent of the choice of q. However, if the dimension of the non-CM locus is larger than zero, little is known about how this dimension is related to the structure of the local ring A . The purpose of this paper is to show that if M is a finitely generated A -module, then there exist systems of parameters x = ( x 1 , …, x d ) (where d = dim M ) such that the difference is a polynomial in n 1 , …, n d for all positive integers n 1 , …, n d and the degree of I M ( n 1 , …, n d ;x ) is independent of the choice of x . We shall also give various characterizations of this degree by using the notion of reducing systems of parameters of Auslander and Buchsbaum[l]. In particular, if the module M is equidimensional we shall show that the degree of I M ( n 1 , …, n d ;x ) is equal to the dimension of the non-CM locus of M .

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.

On the length of the powers of systems of parameters in local ring

Throughout this note, A denotes a commutative local Noetherian ring with maximal ideal m and M a finitely generated A -module with dim ( M ) = d . Let x 1 , …, x d be a system of parameters (s.o.p. for short) for M and I the ideal of A generated by x 1 , …, x d .

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.

ON A NEW INVARIANT OF FINITELY GENERATED MODULES OVER LOCAL RINGS

Let

On the cofiniteness of generalized local cohomology modules

M

Asymptotic stability of certain sets of associated prime ideals of local cohomology modules

be a finitely generated module on a local ring