Graded components of Local cohomology modules II

Let $A$ be a commutative Noetherian ring containing a field $K$ of characteristic zero and let $R= A[X_1, \ldots, X_m]$. Consider $R$ as standard graded with $\deg A=0$ and $\deg X_i=1$ for all $i$. We present a few results about the behavior of the graded components of local cohomology modules $H_I^i(R)$ where $I$ is an arbitrary homogeneous ideal in $R$. We mostly restrict our attention to the Vanishing, Tameness and Rigidity problems.