Modules cofinite and weakly cofinite with respect to an ideal

The purpose of the present paper is to continue the study of modules cofinite and weakly cofinite with respect to an ideal $\frak a$ of a Noetherian ring $R$. It is shown that an $R$-module $M$ is cofinite with respect to $\frak a$, if and only if, \Ext^i_R(R/\frak a,M) is finitely generated for all $i\leq {\rm cd}(\frak a,M)+1$, whenever $\dim R/\frak a=1$. In addition, we show that if $M$ is finitely generated and $H^i_{\frak a}(M)$ are weakly Laskerian for all $i\leq t-1$, then $H^i_{\frak a}(M)$ are ${\frak a}$-cofinite for all $i\leq t-1$ and for any minimax submodule $K$ of $H^{t}_{\frak a}(M)$, the $R$-modules \Hom_R(R/{\frak a}, H^{t}_{\frak a}(M)/K) and \Ext^{1}_R(R/{\frak a}, H^{t}_{\frak a}(M)/K) are finitely generated, where $t$ is a non-negative integer. Finally, we explore a criterion for weakly cofiniteness of modules with respect to an ideal of dimension one. Namely for such ideals it suffices that the two first \Ext-modules in the definition for weakly cofiniteness are weakly Laskerian. As an application of this result we deduce that the category of all ${\frak a}$-weakly cofinite modules over $R$ forms a full Abelian subcategory of the category of modules.Comment: 15 pages, To appear in J. Algebra Appl. arXiv admin note: text overlap with arXiv:1308.604

Cofiniteness of weakly Laskerian local cohomology modules

Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules $T$ with $\dim T\leq n$ and we show it by ${\rm FD_{\leq n}}$ where $n\geq -1$ is an integer. We prove that for any ${\rm FD_{\leq 0}}$(or minimax) submodule N of $H^t_I(M)$ the R-modules ${\rm Hom}_R(R/I,H^{t}_I(M)/N) {\rm and} {\rm Ext}^1_R(R/I,H^{t}_I(M)/N)$ are finitely generated, whenever the modules $H^0_I(M)$, $H^1_I(M)$, ..., $H^{t-1}_I(M)$ are ${\rm FD_{\leq 1}}$ (or weakly Laskerian). As a consequence, it follows that the associated primes of $H^{t}_I(M)/N$ are finite. This generalizes the main results of Bahmanpour and Naghipour, Brodmann and Lashgari, Khashyarmanesh and Salarian, and Hong Quy. We also show that the category $\mathscr {FD}^1(R,I)_{cof}$ of $I$-cofinite ${\rm FD_{\leq1}}$ ~ $R$-modules forms an Abelian subcategory of the category of all $R$-modules.Comment: 8 pages, some changes has been don

On the annihilators of local cohomology modules

AbstractLet (R,m) be a commutative Noetherian complete local ring, M a non-zero finitely generated R-module of dimension d⩾1, and TR(M):=⋃{N:N⩽M and dimN<dimM}. In this paper we calculate the annihilator of the top local cohomology module Hmd(M). More precisely, we show that 0:RHmd(M)=0:RM/TR(M). Moreover, for every positive integer n, we calculate the radical of the annihilator of Hmn(M). More precisely, we prove that if Hmn(M) is not finitely generated then Rad(0:RHmn(M))=⋂p∈Sp, whereS={p∈SpecR:(Hpn−1(M))p≠0 and dimR/p=1}