Graded components of local cohomology modules of $\mathfrak{C}$-monomial ideals in characteristic zero

TL;DR

This paper investigates the structure and invariants of graded components of local cohomology modules supported on $rak{C}$-monomial ideals over characteristic zero rings, revealing their dependence on negative coordinates and providing bounds and structure theorems.

Contribution

It extends the understanding of local cohomology components for $rak{C}$-monomial ideals, including invariants like Bass numbers and injective dimensions, in a general setting.

Findings

Components depend only on negative coordinates of the grading vector.

Finiteness of Bass numbers is established under regularity assumptions.

A structure theorem is provided for components over power series rings.

Abstract

Let $A$ be a commutative Noetherian ring of characteristic zero and $R=A[X_1, \ldots, X_d]$ be a polynomial ring over $A$ with the standard $\mathbb{N}^d$-grading. Let $I\subseteq R$ be an ideal which can be generated by elements of the form $aU$ where $a \in A$ (possibly nonunit) and $U$ is a monomial in $X_i$'s. We call such an ideal as a `$\mathfrak{C}$-monomial ideal'. Local cohomology modules supported on monomial ideals gain a great deal of interest due to their applications in the context of toric varieties. It was observed that for $\underline{u} \in \mathbb{Z}^d$, their $\underline{u}^{th}$ components depend only on which coordinates of $\underline{u}$ are negative. In this article, we show that this statement holds true in our general setting, even for certain invariants of the components. We mainly focus on the Bass numbers, injective dimensions, dimensions, associated primes, Bernstein-type dimensions, and multiplicities of the components. Under the extra assumption that $A$ is regular, we describe the finiteness of Bass numbers of each component and bound its injective dimension by the dimension of its support. Finally, we present a structure theorem for the components when $A$ is the ring of formal power series in one variable over a characteristic zero field.