Graded components of local cohomology modules over polynomial rings

Abstract

Let $K$ be a field and let $R=K[X_1,\dots ,X_m]$ with $m\ge 2$. Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1\le g\le m-1$. Suppose $H_I^i(R)\ne 0$ for some $i\ge 0$. We show (1) $H_I^i(R{)}_n\ne 0$ for all $n\le -m$. (2) if Supp $H_I^i(R)\ne \{(X_1,\dots ,X_m)\}$ then $H_I^i(R{)}_n\ne 0$ 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) ${\dim }_K{H}_I^g(R{)}_n$ 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