For a smooth scheme over a field the sheaf of crystalline differential operators is the the βenveloping algebroid of the tangent Lie algebroidβ of : to an affine it assigns the algebra that is generated over from the -module of vector fields (derivations of ), subject to the relations
and
for all and .
If the field has characteristic 0 this is the ordinary sheaf of differential operators
for instance section 3.1 of
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