Graded components of local cohomology modules supported C-monomial ideals

Abstract

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[X1,...,Xn] be a polynomial ring and I = (a1U1,...,acUc) ⊆ R an ideal, where aj ∈ A (not necessarily units) and Uj ’s are monomials in X1,...,Xn. 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 Hi 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