Graded components of local cohomology modules over polynomial rings

TL;DR

This paper investigates the structure of graded components of local cohomology modules over polynomial rings, revealing non-vanishing patterns and infinite dimensions under certain conditions, with implications for Lyubeznik functors.

Contribution

It provides new results on the non-vanishing and infinite dimensionality of graded local cohomology modules over polynomial rings, extending understanding of their structure.

Findings

Non-vanishing of graded components for degrees ≤ -m.

Infinite dimensionality of certain graded components in characteristic zero.

Results apply to a broad class of Lyubeznik functors.

Abstract

Let $K$ be a field and let $R = K[X_1, \ldots, X_m]$ with $m \geq 2$. Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1 \leq g \leq m -1$. Suppose $H^i_I(R) \neq 0$ for some $i \geq 0$. We show (1) $H^i_I(R)_n \neq 0$ for all $n \leq -m$. (2) if Supp $H^i_I(R) \neq \{ (X_1, \ldots, X_m)\}$ then $H^i_I(R)_n \neq 0$ for all $n \in \mathbb{Z}$. Furthermore if char $K = 0$ then $\dim_K H^i_I(R)_n$ is infinite for all $n \in \mathbb{Z}$. (3) $\dim_K H^g_I(R)_n$ is infinite for all $n \in \mathbb{Z}$. In fact we prove our results for $\mathcal{T}(R)$ where $\mathcal{T}(-)$ is a large sub class of graded Lyubeznik functors