General Local Cohomology Modules with Small Dimensions

Let Φ be a system of ideals of a noetherian ring R and M a finitely generated R -module. We introduce the notion n -depth(Φ, M ) and show that inf { i | dim Supp R H Φ i ( M ) > n } = n -depth ( Φ , M ) , n = − 1,0,1. If ( R , m ) is a complete local ring, n is a non negative integer and M is a finitely generated R -module such that dim Supp R H Φ i ( M ) ≤ 2 for all i < n , then H Φ i ( M ) is Φ-weakly cofinite for all i < n .