Zero-dimensional Non-Artinian local cohomology modules

Abstract

Let $(R,{\mathfrak {m}},k)$ be a Noetherian local ring of dimension $d\ge 4$. Assume that $2\le i \le d-2$ is an integer and $x_1,\ldots ,x_i$ is a part of a system of parameters for R. Let $\Upsilon _i$ denote the set of all prime ideals ${\mathfrak {p}}$ of R such that $\dim R/{\mathfrak {p}}=i+1$, ${\text {Supp}}H^i_{(x_1,\ldots ,x_i)R}(R/{\mathfrak {p}})=\{{\mathfrak {m}}\}$, and $\dim _{k} {\text {Soc}}_R H^i_{(x_1,\ldots ,x_i)R}(R/{\mathfrak {p}})=\infty$. In this paper, it is shown that $\Upsilon _i$ is an infinite set.