Abstract:
Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $M$ be a finitely generated $R$-module and $\mathfrak{a}$, $I$ and $J$ be ideals of $R$. We investigate the structure of formal local cohomology modules of $\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},I,J}(M)$ with respect to a pair of ideals, for all $i\geq 0$. The main subject of the paper is to study the finiteness properties and Artinianness of $\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},\mathfrak{m},J}(M)$. We study the maximum and minimum integer $i\in \N$ such that $\mathfrak{F}^i_{\mathfrak{a},\mathfrak{m},J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},\mathfrak{m},J}(M)$ are not Artinian. We obtain some results involving cossuport, coassociated and attached primes for formal local cohomology modules with respect to a pair of ideals. Also, we give an criterion involving the concepts of finiteness and vanishing of formal local cohomology modules and \v{C}ech-formal local cohomology modules with respect to a pair of ideals.

Abstract:
Let $R$ be a noetherian ring, $\fa$ an ideal of $R$ such that $\dim R/\fa=1$ and $M$ a finite $R$--module. We will study cofiniteness and some other properties of the local cohomology modules $\lc^{i}_{\fa}(M)$. For an arbitrary ideal $\fa$ and an $R$--module $M$ (not necessarily finite), we will characterize $\fa$--cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.

Abstract:
Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. In this paper we introduce some criterions on Artinianness of formal local cohomology, in particular vanishing and finiteness of local cohomology modules. We find out the lower and upper bound for Artinianness of formal local cohomology

Abstract:
Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, $M$ an $R$--module and $n$ a non-negative integer. In this paper we first will study the finiteness properties of the kernel and the cokernel of the natural map $f:\Ext^n_{R}(R/\fa,M)\lo \Hom_{R}(R/\fa,\lc^{n}_{\fa}(M))$. Then we will get some corollaries about the associated primes and artinianness of local cohomology modules. Finally we will study the asymptotic behaviour of the kernel and the cokernel of this natural map in the graded case.

Abstract:
In continuation of [1] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Z\"oschinger on coassociated primes of arbitrary modules with results from [1], [4], [5], [6] and obtain partial answers to questions which were left open in [1]. These partial answers give further support for conjecture $(*)$ from [1] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Z\"oschinger, we prove some non-finiteness results of local cohomology.

Abstract:
Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.

Abstract:
This paper centers around Artinianness of the local cohomology of $ZD$-modules. Let $\fa$ be an ideal of a commutative Noetherian ring $R$. The notion of $\fa$-relative Goldie dimension of an $R$-module $M$, as a generalization of that of Goldie dimension is presented. Let $M$ be a $ZD$-module such that $\fa$-relative Goldie dimension of any quotient of $M$ is finite. It is shown that if $\dim R/\fa=0$, then the local cohomology modules $H^i_{\fa}(M)$ are Artinian. Also, it is proved that if $d=\dim M$ is finite, then $H^d_{\fa}(M)$ is Artinian, for any ideal $\fa$ of $R$ . These results extend the previously known results concerning Artinianness of local cohomology of finitely generated modules.

Abstract:
Colocalization is a right adjoint to the inclusion of a subcategory. Given a ring-spectrum R, one would like a spectral sequence which connects a given colocalization in the derived category of R-modules and an appropriate colocalization in the derived category of graded modules over the graded ring of homotopy groups of R. We show that, under suitable conditions, such a spectral sequence exists. This generalizes Greenlees' local-cohomology spectral sequence. The colocalization spectral sequence introduced here is associated with a localization spectral sequence, which is shown to be universal in an appropriate sense. We apply the colocalization spectral sequence to the cochains of certain loop spaces, yielding a non-commutative local-cohomology spectral sequence converging to the shifted cohomology of the loop space, a result dual to the local-cohomology theorem of Dwyer, Greenlees and Iyengar. An application to the abutment term of the Eilenberg-Moore spectral sequence is also presented.

Abstract:
A general model for the early recognition and colocalization of homologous DNA sequences is proposed. We show, on a thermodynamic ground, how the distance between two homologous DNA sequences is spontaneously regulated by the concentration and affinity of diffusible mediators binding them, which act as a switch between two phases corresponding to independence or colocalization of pairing regions.

Abstract:
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax $R$-module $M$ of krull dimension less than 3, with respect to $\mathcal{S}$. There are some results on cofiniteness and artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module $M$ of finite krull dimension and an integer $n\in\mathbb{N}$, if $\lc^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then $\lc^{i}_{I,J}(M)/\fa^{j}\lc^{i}_{I,J}(M)\in\mathcal{S}$ for any $\fa\in\tilde{W}(I,J)$, all $i\geq n$, and all $j\geq0$. By introducing the concept of Seree cohomological dimension of $M$ with respect to $(I,J)$, for an integer $r\in\mathbb{N}_0$, $\lc^{j}_{I,J}(R)\in\mathcal{S}$ for all $j>r$ iff $\lc^{j}_{I,J}(M)\in\mathcal{S}$ for all $j>r$ and any finite $R$-module $M$.