A note on Azumaya algebras and one-forms

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- 1 cover of X ′ on which the Azumaya algebra does not split, answering a question of Sasha Petrov.