Graded components of local cohomology modules supported on $\mathfrak{C}$-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 $R=A[X_1,\dots ,X_n]$ be a polynomial ring and $I=(a_1{U}_1,\dots ,a_c{U}_c)\subseteq R$ an ideal, where $a_j\in A$ (not necessarily units) and $U_j$'s are monomials in $X_1,\dots ,X_n$. 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 $i\ge 0$. 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.

20 pages