U Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a e et td d – – K Ky yo oz za ai ir re e, , J J K K ((2 20 00 05 5)) i DEDICATED to My husband, John U Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a e et td d – – K Ky yo oz za ai ir re e, , J J K K ((2 20 00 05 5)) ii ACKNOWLEDGEMENTS I acknowledge the following people for the unreserved support they rendered. Professo...

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Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standardgraded ring, $M$ be a finitely generated graded $R$-module and $J$be a homogenous ideal of $R$. In this paper we study the gradedstructure of the $i$-th local cohomology module of $M$ defined by apair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. Moreprecisely, we discuss finiteness property and vanishing of thegraded components $H^...