On the vanishing and finiteness properties of generalized local cohomology modules

Let $R$ be a commutative noetherian ring, $\text{fa}$ an ideal of $R$ and $M,N$ finite $R$–modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] ${\text{lc}}_{\text{fa}}^i(M,N)$ is finite for all $i<n$. \item[(ii)] ${\text{Coass}}_R({\text{lc}}_{\text{fa}}^i(M,N))\subset \text{V}(\text{fa})$ for all $i<n$. \item[(iii)] ${\text{lc}}_{\text{fa}}^i(M,N)$ is coatomic for all $i<n$. \end{enumerate} If $\text{pd}M$ is finite and $r$ be a non-negative integer such that $r>\text{pd}M$ and ${\text{lc}}_{\text{fa}}^i(M,N)$ is finite (resp. minimax) for all $i\ge r$, then ${\text{lc}}_{\text{fa}}^i(M,N)$ is zero (resp. artinian) for all $i\ge r$.