The Noetherian Properties of the Rings of Differential Operators on Central 2-Arrangements

Abstract

Whereas Holm proved that the ring of differential operators on a generic hyperplane arrangement is finitely generated as an algebra, the problem of its Noetherian properties is still open. In this article, after proving that the ring of differential operators on a central arrangement is right Noetherian if and only if it is left Noetherian, we prove that the ring of differential operators on a central 2-arrangement is Noetherian. In addition, we prove that its graded ring associated to the order filtration is not Noetherian when the number of the consistuent hyperplanes is greater than 1.

Key Words:

• Hyperplane arrangement
• Noetherian property
• Ring of differential operators

2010 Mathematics Subject Classification:

• Primary 13N10
• Secondary 32S22

Notes

Communicated by A. Smoktunowicz.