The noetherian properties of the rings of differential operators on central 2-arrangements

TL;DR

This paper investigates the Noetherian properties of rings of differential operators on central 2-arrangements, establishing conditions for their Noetherianity and analyzing associated graded rings.

Contribution

It proves that the ring of differential operators on a central 2-arrangement is Noetherian and clarifies the equivalence of right and left Noetherianity for these rings.

Findings

The ring is right Noetherian if and only if it is left Noetherian.

The ring of differential operators on a central 2-arrangement is Noetherian.

The associated graded ring is not Noetherian when the number of hyperplanes exceeds one.

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.