On the vanishing, artinianness and finiteness of local cohomology modules

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global Principle for minimax local cohomology modules is shown. If $\LC^{i}_{\fa}(M)$ is coatomic for $i\leq r$ ($M$ finite) then $\LC^{i}_{\fa}(M)$ is finite for $i\leq r$. 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.