  Let $A$ be a Dedekind domain of characteristic zero such that its
localization at every maximal ideal has mixed characteristic with finite
residue field. Let $R=A[X_1,\ldots, X_n]$ be a polynomial ring and $I=(a_1U_1,
\ldots, a_c U_c)\subseteq R$ an ideal, where $a_j \in A$ (not necessarily
units) and $U_j$'s are monomials in $X_1, \ldots, X_n$. We call such an ideal
$I$ as a $\mathfrak{C}$-monomial ideal. Consider the standard multigrading on
$R$. We produce a structure theorem for the multigraded components of the local
cohomology modules $H^i_I(R)$ for $i \geq 0$. We further analyze the torsion
part and the torsion-free part of these components. We show that if $A$ is a
PID then each component can be written as a direct sum of its torsion part and
torsion-free part. As a consequence, we obtain that their Bass numbers are
finite.
