Generalization of an example of Hartshorne concerning local cohomology

TL;DR

This paper generalizes a Hartshorne example showing that certain local cohomology modules are not artinian, specifically in a formal power series ring over a field with a particular ideal and prime element.

Contribution

It extends Hartshorne's example to a broader setting, demonstrating non-artinianness of local cohomology modules under more general conditions.

Findings

H^{n-2}_I (R/pR) is not artinian in the generalized setting

The result applies to formal power series rings over fields with specific ideals

Provides insight into the structure of local cohomology modules in algebraic geometry

Abstract

We prove the following generalization of an example of Hartshorne: Let k be a field, n=4, R = k[[X1,...,Xn]], I = (X1,...,Xn-2)R and p\in R a prime element such that p\in(Xn-1,Xn)R. Then H^{n-2}_I (R/pR) is not artinian.