Generalized Formal Local Cohomology Modules

Let a be an ideal of a local ring $(R, m)$ and M and N two finitely generated R-modules. In this paper, we introduce the concept of generalized formal local cohomology modules. We define i-th generalized formal local cohomology module of M and N with respect to a by Fₐⁱ(M,N) := nHₘⁱ(M,N/aⁿN ) for i≥ 0. We prove several results concerning vanishing and finiteness properties of these modules.