On the annihilation of local cohomology modules

Let R be a (not necessary finite dimensional) commutative noetherian ring and let C be a semi-dualizing module over R. There is a generalized Gorenstein dimension with respect to C, namely G(C)-dimension, sharing the nice properties of Auslander's Gorenstein dimension. In this paper, we establish the Faltings' Annihilator Theorem and it's uniform version (in the sense of Raghavan) for local cohomology modules over the class of finitely generated R-modules of finite G(C)-dimension, provided R is Cohen-Macaulay. Our version contains variations of results already known on the Annihilator Theorem.