  Let $A$ be a regular ring containing a field 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 $i$. In this paper we present a comprehensive study
of graded components of local cohomology modules $H^i_I(R)$ where $I$ is an
\emph{arbitrary} homogeneous ideal in $R$. Our study seems to be the first in
this regard.
