On Endomorphism Rings of Local Cohomology Modules

Let I be an ideal of a Complete Cohen-Macaulay local ring R of dimension n. We wil show that the natural homomorphism Rto HomR(HcI(KR), HcI(KR)) is an isomorphism provided that I is a cohomologically compltete intersection ideal of grade c where KR (resp. HiI(.)) denote the canonical module (resp. i-th local cohomology with respect to the ideal I) of R. The same result is true for the Matlis dual of HcI(KR).Comment: Accepted in International Journal of Algebra; International Journal of Algebra(2013

On generalized completion homology modules

Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be any $R$-modules. We define the generalized completion homology modules $L_i\Lambda^I (N,M)$, for $i\in \mathbb{Z}$, as the homologies of the complex $\lim\limits_{\longleftarrow}(N/I^sN\otimes_R F_{\cdot}^R)$. Here $F_{\cdot}^R$ denote a flat resolution of $M$. In this article we will prove the vanishing and non-vanishing properties of $L_i\Lambda^I (N,M)$. We denote $H^{i}_{I}(N,M)$ (resp. $U^I_i(N,M)$) by the generalized local cohomology modules (resp. the generalized local homology modules). As a technical tool we will construct several natural homomorphisms of $L_i\Lambda^I (N,M)$, $H^{i}_{I}(N,M)$ and $U^I_i(N,M)$. We will investigate when these natural homomorphisms are isomorphisms. Moreover if $M$ is Artinian and $N$ is finitely generated then it is proven that $L_i\Lambda^I (N,M)$ is isomorphic to $U^I_i(N,M)$ for each $i\in \mathbb{Z}$. The similar result is obtained for $H^i_{I}(N,M)$. Furthermore if both $M$ and $N$ are finitely generated with c=\grade(I,M). Then we are able to prove several necessary and sufficient conditions such that $H^i_{I}(M)=0$ for all $i\neq c.$ Here $H^i_{I}(M)$ denote the ordinary local cohomology module.Comment: 17 page

On natural homomorphisms of local cohomology modules

Let $M$ be a non-zero finitely generated module over a finite dimensional commutative Noetherian local ring $(R,\mathfrak{m})$ with dim$_R(M)=t$. Let $I$ be an ideal of $R$ with grade$(I,M)=c$. In this article we will investigate several natural homomorphisms of local cohomology modules. The main purpose of this article is to investigate that the natural homomorphisms Tor$^R_c(k,H^c_I(M))\to k\otimes_R M$ and Ext$^{d}_R(k,H^c_I(M))\to {\rm Ext}^t_R(k, M)$ are non-zero where $d:=t-c$. In fact for a Cohen-Macaulay module $M$ we will show that the homomorphism Ext$^d_R(k,H^c_I(M))\to {\rm Ext}^t_R(k, M)$ is injective (resp. surjective) if and only if the homomorphism $H^{d}_{\mathfrak{m}}(H^c_{I}(M))\to H^t_{\mathfrak{m}}(M)$ is injective (resp. surjective) under the additional assumption of vanishing of Ext modules. The similar results are obtained for the homomorphism Tor$^R_c(k,H^c_I(M))\to k\otimes_R M$. Moreover we will construct the natural homomorphism ${\rm Tor}^R_c(k, H^c_I(M))\to {\rm Tor}^R_c(k, H^c_J(M))$ for the ideals $J\subseteq I$ with $c = {\rm grade}(I,M)= {\rm grade}(J,M)$. There are several sufficient conditions on $I$ and $J$ to prove this homomorphism is an isomorphism.Comment: submitted, keyword: Commutative Algebr

Cohomologically complete intersections with vanishing of Betti numbers

Let $I$ be ideal of an $n$-dimensional local Gorenstein ring $R$. In this paper we will describe several necessary and sufficient conditions such that the ideal $I$ becomes cohomologically complete intersections. In fact, as a technical tool, it will be shown that the vanishing $H^i_{I}(R)= 0$ for all i\neq c= \grade (I) is equivalent to the vanishing of the Betti numbers of $H^c_{I}(R)$. This gives a new characterization to check the cohomologically complete intersections property with the homological properties of the vanishing of Tor modules of $H^c_{I}(R)$.Comment: submitte

A Few Comments On Matlis Duality

For a Noetherian local ring $(R,{\mathfrak m})$ with \mathfrak p\in \Spec(R) we denote $E_R(R/\mathfrak p)$ by the $R$-injective hull of $R/\mathfrak p$. We will show that it has an $\hat{R}^\mathfrak p$-module structure and there is an isomorphism $E_R(R/\mathfrak p)\cong E_{\hat{R}^\mathfrak p}(\hat{R}^\mathfrak p/\mathfrak p\hat{R}^\mathfrak p)$ where $\hat{R}^\mathfrak p$ stands for the $\mathfrak p$-adic completion of $R$. Moreover for a complete Cohen-Macaulay ring $R$ the module $D(E_R(R/\mathfrak p))$ is isomorphic to $\hat{R}_\mathfrak{p}$ provided that $\dim(R/\mathfrak p)=1$ and $D(\cdot)$ denotes the Matlis dual functor \Hom_R(\cdot, E_R(R/\mathfrak m)). Here $\hat{R}_\mathfrak{p}$ denotes the completion of ${R_\mathfrak p}$ with respect to the maximal ideal $\mathfrak pR_\mathfrak p$. These results extend those of Matlis (see \cite{m}) shown in the case of the maximal ideal ${\mathfrak m}$.Comment: 9 pages,to be appeared in International Electronic Journal of Algebr

On Cohomologically Complete Intersections in Cohen-Macaulay Rings

An ideal I of a local Cohen-Macaulay ring R is called a cohomologically complete intersection if H^i_I(R) = 0 for all i \neq c = height(I). Here H^i_I(R), i \in Z denotes the local cohomology of R with respect to I. For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view. As a main result it is shown that the vanishing H^iI_(M) = 0 for all i \neq c is completely encoded in homological properties of H^cI_(M). These results extend those of Hellus and Schenzel (see [13, Theorem 0.1]) shown in the case of a local Gorenstein ring. In particular we get a characterization of cohomologically complete intersections in a Cohen-Macaulay ring in terms of the canonical module.Comment: 16 pages, submitted. arXiv admin note: text overlap with arXiv:0804.2558 by other author

Roughness in quotient groups

The theory of rough sets was firstly introduced by Pawlak (see \cite{p}). Many Mathematician has been studied the relations between rough sets and algebraic systems such as groups, rings and modules. In this paper we will introduce the lower and upper approximations in a quotient group. We will discuss several properties of the lower and upper approximations. Moreover under some additional assumptions we are able to show that the lower approximation is a normal subgroup of the quotient group but this property fails for the upper approximation. At the end we will develop several homomorphisms between lower approximations.Comment: sumitte

The Double Jones Birefringence in Magneto-electric Medium

In this paper, the Maxwell's equations for the tensorial magneto-electric (ME) medium have been solved which in fact is the extension of anisotropic nonmagnetic medium. All of the dielectric permittivity, magnetic permeability and the ME tensors are considered. The transverse polarization is shown explicitly and the propagation of electromagnetic wave in the ME medium is found to have the Double Jones Birefringence. We also find the condition of D'yakonov surface wave for magneto-isotropic but with ME anisotropic medium. Especially when the incident angle is $\frac{\pi}{4}$, it may be measurable in principle

Photometric study of contact binary star MW And

The Tarleton Observatory's 0.8m telescope and CCD photometer were used to obtain 1298 observations of the short period eclipsing binary star MW And. The observations were obtained in Johnson's BVR filters. The light curves show that MW And is an eclipsing binary star with a period of 0.26376886 days. Further analysis showed that the period of MW And is changing at the rate of 0.17sec/year. The photometric solutions were obtained using the 2015 version of the Wilson-Devinney model. The solutions show that MW And is an eclipsing binary star of W UMa type. Our analysis suggests that the system has a light curve of W-subtype contact system. Its spectral type of K0/K1, as estimated from its color, places it in the Zero-Age contact zone of the period-spectral diagram. Luminosity from the solutions indicate that it is a double-line spectroscopic system and therefore, spectroscopic observations are recommended for further detail study

On Invariants and Endomorphism Rings of Local Cohomology Modules

Let $(R,\mathfrak{m})$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ with \grade I = c we define new numerical invariants $\tau_{i,j}(I)$ as the socle dimensions of $H^i_{\mathfrak{m}}(H^{n-j}_I(R))$. In case of a regular local ring containing a field these numbers coincide with the Lyubeznik numbers $\lambda_{i,j}(R/I)$. We use $\tau_{d,d}(I), d = \dim R/I,$ to characterize the surjectivity of the natural homomorphism f : \hat{R} \to \Hom_{\hat{R}}(H^c_{I\hat{R}}(\hat{R}),H^c_{I\hat{R}}(\hat{R})). As a technical tool we study several natural homomorphisms. Moreover we prove a few results on $\tau_{i,j}(I)$.Comment: Accepted in Journal of Algebr