Graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals

TL;DR

This paper provides a detailed structure theorem for the multigraded components of local cohomology modules supported on $rak{C}$-monomial ideals over Dedekind domains, analyzing torsion and torsion-free parts and establishing finiteness of Bass numbers.

Contribution

It introduces a structure theorem for multigraded local cohomology components supported on $rak{C}$-monomial ideals over Dedekind domains, including torsion analysis and Bass number finiteness.

Findings

Components decompose into torsion and torsion-free parts over PIDs.

Bass numbers of these components are finite.

Structure theorem applies to multigraded components of local cohomology modules.

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 $R=A[X_1,\ldots, X_n]$ 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.