On the annihilators and attached primes of top local cohomology modules

Abstract

Abstract. Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that AnnR(

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:…”

Annihilators of local homology modules

“…There are many results about annihilators of local cohomology modules. For example, the following theorem is a main result of [1] on the annihilators of top local cohomology modules. Here, as the dual case of the above result, we obtain some results about the annihilator of top local homology modules.…”

“…In this paper we study the top local homology module H a n (M ), where M is a nonzero Artinian R-module of Noetherian dimension n and a is an arbitrary ideal of R. The module H a n (M ) is called a top local homology module because max{i : H a i (M ) = 0} n by [5], Proposition 4.8. The problem of finding annihilators of local cohomology modules has been studied by several authors; see for example [1], [2] and [3]. In [3], the authors proved that if (R, m) is a complete Noetherian local ring and M is a finitely generated R-module then Ann R (H dim M m (M )) = T R (M ), where T R (M ) is the largest submodule of M such that dim T R (M ) < dim(M ).…”

“…[1], Theorem 2.3). Let R be a Noetherian ring and a an ideal of R. Let M be a nonzero finitely generated R-module such that cd(a,M ) = dim M .…”

Some results on the annihilators and attached primes of local cohomology modules

“…Another main result in Section 3 is to give a complete characterization of the attached primes of the local cohomology module H dim M −1 a (M). More precisely, we shall show the following result, which is an extension of the main theorems of [3] and [9]. Throughout this paper, R will always be a commutative Noetherian ring with non-zero identity and a will be an ideal of R. For each R-module L, we denote by Assh R L (resp.…”