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crystalline differential operator

Definition

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

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

and

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

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

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

Related concepts

• crystalline site, crystalline cohomology

• D-module

References

for instance section 3.1 of

• Roman Bezrukavnikov, Noncommutative Counterparts of the Springer Resolution (pdf)