For a smooth scheme over a field the sheaf of crystalline differential operators is the the “enveloping algebroid of the tangent Lie algebroid” of : to an affine it assigns the algebra that is generated over from the -module of vector fields (derivations of ), subject to the relations
v_1 \cdot v_2 - v_2 \cdot v_1 = [v_1, v_2]
v_1 \cdot f - f \cdot v_1 = v_1(f)
for all and .
If the field has characteristic 0 this is the ordinary sheaf of differential operators
for instance section 3.1 of