Graded components of local cohomology modules supported on C-monomial ideals

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<inf>1</inf>,…,X<inf>n</inf>] be a polynomial ring and I=(a<inf>1</inf>U<inf>1</inf>,…,a<inf>c</inf>U<inf>c</inf>)⊆R an ideal, where a<inf>j</inf>∈A (not necessarily units) and U<inf>j</inf>'s are monomials in X<inf>1</inf>,…,X<inf>n</inf>. We call such an ideal I as a C-monomial ideal. Consider the standard multigrading on R. We produce a structure theorem for the multigraded components of the local cohomology modules H<inf>I</inf>ⁱ(R) for i≥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.