The Ring of Differential Operators on a Nodal Curve is not a Bialgebroid

TL;DR

This paper demonstrates that the ring of differential operators on a nodal curve does not possess a bialgebroid structure, contrasting with cases where local projectivity ensures such a structure.

Contribution

It provides an elementary proof that the differential operators on a nodal curve lack local projectivity and bialgebroid structure, challenging previous assumptions.

Findings

The ring of differential operators on a nodal curve is not locally projective.

Such a ring does not admit a bialgebroid structure.

Elementary methods suffice to establish these properties.

Abstract

In a previous article, we showed that local projectivity is a sufficient condition for the existence of a bialgebroid structure on the ring of differential operators on an affine variety. In this note, we show using elementary methods that the ring of differential operators on a nodal curve is neither locally projective nor does it admit a bialgebroid structure.