The membership of the local cohomology modules H-a(n) (M) of a module M in certain Serre subcategories of the category of modules is studied from below (i < n) and from above (i > n). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module. (C) 2008 Elsevier Inc. All rights reserved.