Cohomological dimension filtration and annihilators of top local cohomology modules

Abstract

Abstract. Let a denote an ideal in a commutative Noetherian ring R and M a finitely generated R-module. In this paper, we introduce the concept of the cohomological dimen-, where c = cd(a, M ) and M i denotes the largest submodule of M such that cd(a, M i ) ≤ i. Some properties of this filtration are investigated. In particular, in the case that (R, m) is local and c = dim M , we are able to determine the annihilator of the top local cohomology module H

Annihilators of local cohomology modules and simplicity of rings of differential operators

“…In particular, if I is a cohomologically complete interesection ideal (i.e., H i I (R) = 0 for all i = g := grade R (I)), then 0 : R H g I (R) = 0 (see [ If R is unmixed and √ I = m, then 0 : R H d I (R) = 0; the converse holds provided R is, in addition, complete (see [Lyn12,Theorem 2.4 and Corollary 2.5] and [ES12, Theorem 4.2(a)]). (vi) It is worth noting here that there are also several nice results concerning the annihilators of local cohomology modules H i I (M ), where M is a finitely generated R-module; the interested reader may like to consult [BAG12], [ASN14] and [ASN15] for further details. One of the motivations of this paper was to understand a bit better the structure of annihilators of local cohomology modules in mixed characteristic; in particular, in this mixed characteristic setting we have as guide the following:…”

Relative Cohen-Macaulay filtered modules with a view toward relative Cohen-Macaulay modules

“…Atazadeh and et al [1,Theorem 1.1] proved that if a finite R-module M has a cd-filtration, then this filtration is uniquely determined by a reduced primary decomposition of the zero submodule in M. In the present paper, without such a condition on M, we present a cd-filtration for all R-modules whose zero submodule has a primary decomposition (see Corollary 2.28). In section 3, we study relative Cohen-Macaulayness in rings and modules.…”

“…For an R-module M, the cohomological dimension of M with respect to a is defined as cd(a, M) := sup{i ∈ Z | H i a (M) = 0} which is known that for a local ring (R, m) and a = m, this is equal to the dimension of M. For unexplained notation and terminology about local cohomology modules, we refer the reader to [3] and [4]. The notion of cohomological dimension filtration (abbreviated as cd-filtration) of M introduced by A. Atazadeh and et al [1] which is a generalization of the concept of dimension filtration that is defined by P. Schenzel [20] in local case. For any integer 0 ≤ i ≤ cd(a, M), let M i denote the largest submodule of M such that cd(a, M i ) ≤ i.…”

Some bounds for the annihilators of local cohomology and Ext modules

“…Then Bahmanpour [4,Theorem 3.2] extended the result of Lynch for the R-module M . Next, Atazadeh et al [2] proved this equality whenever R is a local ring (not necessarily complete) and finally in [1] they extended it to the non-local case. (Note that T a (M ) = cda(R/pi)=cda(M) M i [2, Remark 2.5], also, if (R, m) is a complete local ring and p ∈ Ass R (M ), then, by the Lichtenbaum-Hartshorne Vanishing Theorem, cd a (R/p) = d if and only if dim R (R/p) = d and √ a + p = m).…”