ABSTRACT Let R be a commutative Noetherian ring, U a system of ideals of R, a 2 U, M an arbitrary R-module and t a non-negative integer. Let S be a Melkersson subcategory of R-modules. Among other things, we prove that if Hi U ðMÞ is in S for all i < t then Hi a ðMÞ is in S for all i < t and for all a 2 U: If S is the class of R-modules N with dimN k where k ! À1, is an inte- ger, then Hi U ðMÞ is in S for all i < t if (and only if) Hi a ðMÞ is in S for all i < t and for all a 2 U: As consequences we study and compare vanishing, Artinianness and support of general local cohomology and ordinary local cohomology supported at ideals of its system of ideals at initial points i < t. We show that SuppR ðHdim U MÀ1 ðMÞÞ is not necessarily finite whenever ðR, mÞ is local and M a finitely generated R-module.
