A note on Azumaya algebras and one-forms

The crystalline differential operators on a smooth variety <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> give rise to a non-split Azumaya algebra over the cotangent bundle of the Frobenius twist <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> . In some cases, this Azumaya algebra splits when restricted to finite covers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> . In this short note, we show that, whenever <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> has a non-closed global one-form, there is a degree- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> cover of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>′</mml:mi> </mml:msup> </mml:math> on which the Azumaya algebra does not split, answering a question of Sasha Petrov.