Cofiniteness Of Local Cohomology Modules Research Articles

Attached and associated primes of local cohomology modules via linkage

Let [Formula: see text] be a commutative Noetherian ring and [Formula: see text] be a finitely generated [Formula: see text]-module. Considering the new concept of linkage of ideals over a module, we study attached and associated prime ideals and cofiniteness of local cohomology modules of [Formula: see text] with respect to some linked ideals over it.

Cofiniteness of local cohomology modules in the class of modules in dimension less than a fixed integer

Let n be a non-negative integer, R a commutative Noetherian ring with dim(R) ≤ n + 2, a an ideal of R, and X an arbitrary R-module...

A study of cofiniteness through minimal associated primes

In this article, we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the well known results in the area.

Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings

Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings

Cofiniteness of local cohomology modules for a pair of ideals for small dimensions

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] and [Formula: see text] be two ideals of [Formula: see text] and [Formula: see text] be an [Formula: see text]-module (not necessary [Formula: see text]-torsion). In this paper among other things, it is shown that if dim [Formula: see text], then the [Formula: see text]-module [Formula: see text] is finitely generated, for all [Formula: see text], if and only if the [Formula: see text]-module [Formula: see text] is finitely generated, for [Formula: see text]. As a consequence, we prove that if [Formula: see text] is finitely generated and [Formula: see text] such that the [Formula: see text]-module [Formula: see text] is [Formula: see text] (or weakly Laskerian) for all [Formula: see text], then [Formula: see text] is [Formula: see text]-cofinite for all [Formula: see text] and for any [Formula: see text] (or minimax) submodule [Formula: see text] of [Formula: see text], the [Formula: see text]-modules [Formula: see text] and [Formula: see text] are finitely generated. Also it is shown that if dim [Formula: see text] (e.g. dim [Formula: see text]) for all [Formula: see text], then the local cohomology module [Formula: see text] is [Formula: see text]-cofinite for all [Formula: see text].

Cofiniteness of Local Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, 𝔪). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and [Formula: see text], then [Formula: see text] is not 𝔭-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then [Formula: see text] is finitely generated if and only if 0 ≤ n ∉ W, where [Formula: see text]. Also, we show that if J ⊆ I are 1-dimensional ideals of R, then [Formula: see text] is J-cominimax, and [Formula: see text] is finitely generated (resp., minimax) if and only if [Formula: see text] is finitely generated for all [Formula: see text] (resp., [Formula: see text]). Moreover, the concept of the J-cofiniteness dimension [Formula: see text] of M relative to I is introduced, and we explore an interrelation between [Formula: see text] and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then [Formula: see text].

Filter Depth and Cofiniteness of Local Cohomology Modules Defined by a Pair of Ideals

Let I, J be ideals of a commutative Noetherian local ring (R, 𝔪) and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that [Formula: see text] is not Artinian. In this paper we show that [Formula: see text] is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that [Formula: see text] is (I,J)-cofinite for all [Formula: see text]. In addition, we show that for a Serre subcategory 𝖲, if [Formula: see text] belongs to 𝖲 for all i > n and if 𝔟 is an ideal of R such that [Formula: see text] belongs to 𝖲, then the module [Formula: see text] belongs to 𝖲.

Some results on the cofiniteness of local cohomology modules

Let R be a commutative Noetherian ring, a an ideal of R, M an R-module and t a non-negative integer. In this paper we show that the class of minimax modules includes the class of AF modules. The main result is that if the R-module ExtRt(R/a,M) is finite (finitely generated), Hai(M) is a-cofinite for all i < t and Hat(M) is minimax then Hat(M) is a-cofinite. As a consequence we show that if M and N are finite R-modules and Hai(N) is minimax for all i < t then the set of associated prime ideals of the generalized local cohomology module Hat(M,N) is finite.

Cofiniteness of local cohomology modules for ideals of small dimension

Let R be a commutative Noetherian ring and let M be a non-zero finitely generated R-module. Let I be an ideal of R and t a non-negative integer such that dim Supp H I i ( M ) ⩽ 1 for all i < t . It is shown that the R-modules H I 0 ( M ) , H I 1 ( M ) , … , H I t − 1 ( M ) are I-cofinite and the R-module Hom R ( R / I , H I t ( M ) ) is finitely generated. This immediately implies that if I has dimension one (i.e., dim R / I = 1 ), then H I i ( M ) is I-cofinite for all i ⩾ 0 . This is a generalization of the main results of Delfino and Marley [D. Delfino, T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997) 45–52] and Yoshida [K.I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997) 179–191] for an arbitrary Noetherian ring R. Also, we prove that if R is local and dim Supp H I i ( M ) ⩽ 2 for all i < t , then the R-modules Ext R j ( R / I , H I i ( M ) ) and Hom R ( R / I , H I t ( M ) ) are weakly Laskerian for all i < t and all j ⩾ 0 . As a consequence, it follows that the set of associated primes of H I i ( M ) is finite for all i ⩾ 0 , whenever dim R / I ⩽ 2 .

Associated primes and cofiniteness of local cohomology modules

Let Open image in new window be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that Open image in new window is finite R-module. If s is the first integer such that the local cohomology module Open image in new window is non Open image in new window-cofinite, then we show that Open image in new window is finite. In particular, the set of associated primes of Open image in new windowis finite. Let Open image in new window be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module Open image in new window is not Open image in new window-cofinite and show that n just depends on the support of M.

Cofiniteness of Local Cohomology Modules Over Regular Local Rings

Cofiniteness of Local Cohomology Modules Over Regular Local Rings

Cofiniteness of Local Cohomology Modules for Principal Ideals

In this note we show that if an ideal I of a noetherian ring is principal, up to radical, then the local cohomology modules with support in V(I) are I-cofinite. 1991 Mathematics Subject Classification 14B15, 13D03, 18G15.

Cofiniteness of local cohomology modules for ideals of dimension one

AbstractIn this paper, we prove that for any ideal I of dimension one is I-cofinite for all i and for any finite A-module M. Furthermore, for any ideal I over any regular local ring A, we investigate the relationship between I-cofiniteness and vanishing for local cohomology modules .