For $X$ a smooth scheme over a field the sheaf$\mathcal{D} = \mathcal{D}_X$ of crystalline differential operators is the the βenveloping algebroid of the tangent Lie algebroidβ of $X$: to an affine $U \to X$ it assigns the algebra that is generated over $\mathcal{O}(U)$ from the $\mathcal{O}(U)$-module$Vect(U)$ of vector fields (derivations of $\mathcal{O}(U)$), subject to the relations

$v_1 \cdot v_2 - v_2 \cdot v_1 = [v_1, v_2]$

and

$v_1 \cdot f - f \cdot v_1 = v_1(f)$

for all $v_1, v_2 \in Vect(U)$ and $f \in \mathcal{O}(U)$.

If the field $k$ has characteristic 0 this is the ordinary sheaf of differential operators