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. The membership of the generalized local cohomology mod-ules H ia (M,N) of two R-modules M and N with respect to an ideal ain certain Serre subcategories of the category of modules is studied frombelow (i<t). Furthermore, the behaviour of the nth graded componentH ia (M,N) n of the generalized local cohomology modules with respect toan ideal containing the irrelevant ideal as n → −∞ is investigated byusing the above result, in certain graded situations. 1. IntroductionGeneralized local cohomology was given in the local case by J. Herzog [7]and in the more general case by Bijan-Zadeh [1]. Let Rbe a commutativeNoetherian ring with identity, aan ideal of Rand let M,Nbe two R-modules.For an integer i≥ 0, the ith generalized local cohomology module H ia (M,N) isdened by H ia (M,N) = lim−→ n∈N Ext iR (M/a n M,N). With M= R, we obtain theordinary local cohomology module H ia (N) of N with respect to a which wasintroduced by Grothendieck.Recall that a class Sof R-modules is a Serre subcategory of the category ofR-modules, when it is closed under taking submodules, quotients and exten-sions. In this paper, we study some properties of generalized local cohomologymodules by using Serre classes.This paper is divided into three sections. In the second section of the paper,we show that if Mis a nitely generated R-module, Nis an arbitrary R-moduleand tis a non-negative integer such that H

FINITENESS PROPERTIES GENERALIZED LOCAL COHOMOLOGY WITH RESPECT TO AN IDEAL CONTAINING THE IRRELEVANT IDEAL