Cofiniteness of General Local Cohomology Modules for Small Dimensions (original) (raw)

On the cofiniteness properties of certain general local cohomology modules

Acta Scientiarum Mathematicarum

Let R be a commutative Noetherian ring, Φ a system of ideals of R, and M a finitely generated R-module. Suppose that a∈Φ and t is a non-negative integer. It is shown that if Ext R i (R/a,H Φ j (M)) is finitely generated for all i and all j<t, then Ext R i (R/a,H Φ t (M)) is finitely generated for i=0,1. In particular, if R is a local ring of dimension at most 2, then Ext R i (R/a,H Φ j (M)) is finitely generated for all i, j.

Cofiniteness of Local Cohomology Modules

Algebra Colloquium, 2014

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 i...

On the cofiniteness of generalized local cohomology modules

Kyoto Journal of Mathematics, 2015

Let R be a commutative Noetherian ring, let I be an ideal of R, and let M , N be two finitely generated R-modules. The aim of this paper is to investigate the I-cofiniteness of generalized local cohomology modules H j I (M, N) = lim − →n Ext j R (M/ I n M, N) of M and N with respect to I. We first prove that if I is a principal ideal, then H j I (M, N) is I-cofinite for all M , N and all j. Secondly, let t be a nonnegative integer such that dim Supp(H j I (M, N)) ≤ 1 for all j < t. Then H j I (M, N) is I-cofinite for all j < t and Hom(R/I, H t I (M, N)) is finitely generated. Finally, we show that if dim(M) ≤ 2 or dim(N) ≤ 2, then H j I (M, N) is I-cofinite for all j. K. I. Kawasaki has proved that if I is a principal ideal in a commutative Noetherian ring, then H j I (N) are I-cofinite for all finitely generated R-modules N and all j ≥ 0 (see [22, Theorem 1]). D. Delfino and T. Marley [12, Theorem 1] and K. I. Yoshida [33, Theorem 1.1] refined result (ii) to more general situation that if

Cofiniteness and non-vanishing of local cohomology modules

Journal of Commutative Algebra, 2014

Let R be a commutative Noetherian local ring, I an ideal of R, and let M be a non-zero finitely generated R-module. In this paper, we establish some new properties of the local cohomology modules H i I (M), i ≥ 0. In particular, we show that if (R, m) is a Noetherian local integral domain of dimension d ≤ 4 which is a homomorphic image of a Cohen-Macaulay ring and x 1 ,. .. , xn is a part of a system of parameters for R, then for all i ≥ 0, the R-modules H i I (R) are I-cofinite, where I = (x 1 ,. .. , xn). Also, we prove that if (R, m) is a Noetherian local ring of dimension d and x 1 ,. .. , xt is a part of a system of parameters for R, then H d−t m (H t (x 1 ,...,xt) (R)) ̸ = 0. In particular, µ d−t (m, H t (x 1 ,...,xt) (R)) ̸ = 0 and injdim R (H t (x 1 ,...,xt) (R)) ≥ d − t. n≥1 Ext i R (R/I n , M). We refer the reader to [4, 6] for more details about local cohomology.

Cofinite Modules and Generalized Local Cohomology

Houston journal of mathematics

Let R be a commutative Noetherian ring, a an ideal of R, and M , N two finitely generated R-modules. We prove that the generalized local cohomology modules H t a (M, N) are a-cofinite; that is, Ext i R (R/a, H t a (M, N)) is finitely generated for all i, t ≥ 0, in the following cases: (i) cd(a) = 1, where cd is the cohomological dimension of a in R. (ii) dim R ≤ 2. Additionally, we show that if cd(a) = 1 then Ext i R (M, H t a (N)) is a-cofinite for all i, t ≥ 0.

Some results on local cohomology modules

Archiv der Mathematik, 2006

Let R be a commutative Noetherian ring, a an ideal of R, and let M be a finitely generated R-module. For a non-negative integer t, we prove that H t a (M) is a-cofinite whenever H t a (M) is Artinian and H i a (M) is a-cofinite for all i < t. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a local ring (R, m), f − depth(a, M) is the least integer i such that H i a (M) ∼ = H i m (M). This result in conjunction with the first one, yields some interesting consequences. Finally, we extend Grothendieck's non-vanishing Theorem to a-cofinite modules.

A generalization of the finiteness problem in local cohomology modules

Proceedings Mathematical Sciences, 2009

Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H i a (N) is a weakly Laskerian R-module for all i < t, then Hom R (R/a, H t a (M, N)) is weakly Laskerian R-module. Also, we prove that Ext i R (R/a, H t a (N)) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp R (H i a (N)) is a finite set for all i < t, then Ext i R (R/a, H t a (N)) is weakly Laskerian R-module for all i = 0, 1.

A generalization of the cofiniteness problem in local cohomology modules

Journal of the Australian Mathematical Society, 2003

Let R be a commutative Noetherian ring with nonzero identity and let M be a finitely generated R-module. In this paper, we prove that if an ideal I of R is generated by a u.s.d-sequence on M then the local cohomology module (M) is I-cofinite. Furthermore, for any system of ideals Φ of R, we study the cofiniteness problem in the context of general local cohomology modules.

On the cofiniteness of local cohomology modules

Mathematical Proceedings of the Cambridge Philosophical Society, 1994

Let (R,m) be a local, noetherian, d-dimensional ring and let M be a finitely generated R-module. Since the local cohomology modules are artinian, is finitely generated for all i and j (see [4], Remark 1*middot;3 and 2·1). Grothendieck[2] made the following conjecture: If I is an ideal of a noetherian ring R, thenis finitely generated for all j.

Cofiniteness of Generalized Local Cohomology Modules

Bulletin of the Australian Mathematical Society, 2011

Let 𝔞 be an ideal of a Noetherian ring R. Let s be a nonnegative integer and let M and N be two R-modules such that ExtjR(M/𝔞M,Hi𝔞(N)) is finite for all i<s and all j≥0 . We show that HomR (R/𝔞,Hs𝔞(M,N)) is finite provided ExtsR(M/𝔞M,N) is a finite R-module. In addition, for finite R-modules M and N, we prove that if Hi𝔞(M,N) is minimax for all i<s, then HomR (R/𝔞,Hs𝔞(M,N)) is finite. These are two generalizations of the result of Brodmann and Lashgari [‘A finiteness result for associated primes of local cohomology modules’, Proc. Amer. Math. Soc. 128 (2000), 2851–2853] and a recent result due to Chu [‘Cofiniteness and finiteness of generalized local cohomology modules’, Bull. Aust. Math. Soc. 80 (2009), 244–250]. We also introduce a generalization of the concept of cofiniteness and recover some results for it.