In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as the Independence theorem, Lichtenbaum-Harshorne vanishing theorem, Basic finiteness and vanishing theorem, among others. In addition, we present a generalized version of the Melkersson theorem regarding the Artinianness of modules, and a result concerning Artinianness of local cohomology modules.
"Graded version of local cohomology with respect to a pair of ideals." J. Commut. Algebra 9 (4) 545 - 561, 2017. https://doi.org/10.1216/JCA-2017-9-4-545