COFINITENESS AND ANNIHILATORS OF TOP LOCAL COHOMOLOGY MODULES

Let R be a Noetherian ring and I be an ideal of R. Let M be a finitely generated R-module with cd(I, M) = t >= 0 and assume that L is the largest submodule of M such that cd(I, L) < cd(I, M). It is shown that Ann(R)H(I)(t) (M) = Ann(R) M/L in each of the following cases: (i) dim M/IM <= 1. (ii) dimR/I <= 1. (iii) The R-module H-I(i)(M) is Artinian for each i >= 2. (iv) The R-module H i I (R) is Artinian for each i >= 2. (v) cd(I, M) <= 1. (vi) cd(I, R) <= 1. (vii) The Rmodule H-I(t)(M) is Artinian and I-cofinite. These assertions answer affirmatively a question raised by Atazadeh et al. in [2], in some special cases.