ON THE COFINITENESS OF IN DIMENSION < 2 LOCAL COHOMOLOGY MODULES FOR A PAIR OF IDEALS

Abstract

In this note, we prove the cofiniteness of local cohomology modules Hi I;J(N) with respect to a pair of ideals (I; J) for all i < t and the finiteness of (0 :Ht I;J (N) I) and Ext1 R(R=I; HI;J t (N)) provided that Exti R(R=I; N) is finitely generated for all i ≤ t + 1 and HI;J i (N) is in dimension < 2 for all i < t, where t ≥ 1 is an integer (here, N is not necessarily finitely generated over R). This extends the results of Bahmanpour-Naghipour [5, Thm 2.6], Bahmanpour-Naghipour-Sedghi [4, Thm 2.8] and H-N [16, Thm 1.1].