serre subcategory, local homology andlocal cohomology

This paper deals with local homology modules and local cohomology modules contained in a serre subcategory of the category of r-modules. for an ideal a of r, we define the concept of the condition c^a on a serre category which is dual to the condition ca of melkersson. as a main result we show that the local homology module ha i (m) of a minimax r-module m of any serre category s satisfying the condition c a belongs to s. also, if s satisfies the condition c^a, then the local cohomology module hi a (m) ∈ s for all i ≥ 0.