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A note on endomorphisms of local cohomology modules
Waqas Mahmood,Zohaib Zahid 대한수학회 2017 대한수학회보 Vol.54 No.1
Let $I$ denote an ideal of a Noetherian local ring $(R,\mathfrak{m})$. Let $M$ denote a finitely generated $R$-module. We study the endomorphism ring of the local cohomology module $H^c_I(M), c = \grade (I,M)$. In particular there is a natural homomorphism $$\Hom_{\hat{R}^I}(\hat{M}^I, \hat{M}^I)\to \Hom_{R}(H^c_{I}(M),H^c_{I}(M)),$$ where $\hat{\cdot}^I$ denotes the $I$-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J \subset I$ with the property $\grade(I,M) = \grade(J,M)$. Our results extends constructions known in the case of $M = R$ (see e.g.~\cite{h1}, \cite{p7}, \cite{p1}).
A NOTE ON ENDOMORPHISMS OF LOCAL COHOMOLOGY MODULES
Mahmood, Waqas,Zahid, Zohaib Korean Mathematical Society 2017 대한수학회보 Vol.54 No.1
Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module $H^c_I(M)$, c = grade(I, M). In particular there is a natural homomorphism $$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$, $where{\hat{\cdot}}^I$ denotes the I-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J{\subset}I$ with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).
Edge version of harmonic index and harmonic polynomial of some classes of graphs
Rabia Nazir,Shoaib Sardar,Sohail Zafar,Zohaib Zahid 한국전산응용수학회 2016 Journal of applied mathematics & informatics Vol.34 No.5
In this paper we define the edge version of harmonic index and harmonic polynomial of a graph $G$. We computed explicit formulas for the edge version of harmonic index and harmonic polynomial of many well known classes of graphs.
EDGE VERSION OF HARMONIC INDEX AND HARMONIC POLYNOMIAL OF SOME CLASSES OF GRAPHS
NAZIR, RABIA,SARDAR, SHOAIB,ZAFAR, SOHAIL,ZAHID, ZOHAIB The Korean Society for Computational and Applied M 2016 Journal of applied mathematics & informatics Vol.34 No.5
In this paper we define the edge version of harmonic index and harmonic polynomial of a graph G. We computed explicit formulas for the edge version of harmonic index and harmonic polynomial of many well known classes of graphs.