A note on Azumaya algebras and one-forms

Abstract

The crystalline differential operators on a smooth variety X give rise to a non-split Azumaya algebra over the cotangent bundle of the Frobenius twist X'. In some cases, this Azumaya algebra splits when restricted to finite covers of X'. In this short note, we show that, whenever X has a non-closed global one-form, there is a degree one cover of X' on which the Azumaya algebra does not split, answering a question of Sasha Petrov.