Finiteness properties of minimax and coatomic local cohomology modules

Let R be a noetherian ring, mathfraka{mathfrak{a}} an ideal of R, and M an R-module. We prove that for a finite module M, if Hⁱ_mathfraka(M){{rm H}^{i}_{mathfrak{a}}(M)} is minimax for all i ≥ r ≥ 1, then Hⁱ_mathfraka(M){{rm H}^{i}_{mathfrak{a}}(M)} is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If Hⁱ_mathfraka(M){{rm H}^{i}_{mathfrak{a}}(M)} is coatomic for i ≤ r (M finite) then Hⁱ_mathfraka(M){{rm H}^{i}_{mathfrak{a}}(M)} is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.