On the minimaxness and coatomicness of local cohomology modules

Abstract

Let R be a commutative Noetherian ring, I an ideal of R and M an R-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and ${\cal C}$ of local cohomology modules. We show that if M is a minimax R-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if n is a nonnegative integer such that (H ⁱ_I (M))_m is a minimax R_m-module for all m ∈ Max(R) and for all i < n, then the set Ass_R(H ⁿ_I (M)) is finite. Also, if H ⁱ_I (M) is minimax for all i ⩾ n ⩾ 1, then H ⁱ_I (M) is Artinian for i ⩾ n. It is shown that if M is a ${\cal C} - {\rm{minimax}}$ module over a local ring such that H ⁱ_I (M) are ${\cal C} - {\rm{minimax}}$ modules for all i < n (or i ⩾ n), where n ⩾ 1, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology modules.