nLab crystalline differential operator

Definition

For $X$ a smooth scheme over a field $k$ the sheaf $\mathcal{D} = \mathcal{D}_X$ 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 $\mathcal{O}(U)$ 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 $f \in \mathcal{O}(U)$.

Characteristic zero

If the field $k$ has characteristic 0, then $\mathcal{D}$ is the ordinary sheaf of differential operators (Berthelot-Ogus, Theroem 2.15).

Positive characteristic

If the field $k$ has positive characteristic, then $\mathcal{D}$ has a large center: the algebra $O_{T^\ast X^{(1)}}$ of functions on the Frobenius twist of the cotangent bundle. Furthermore, $\mathcal{D}$ is Azumaya over its center (Bezrukavnikov-Mirković-Rumynin, Theorem 2.2.3).

Related concepts

• crystalline site, crystalline cohomology

• D-module

References

• P. Berthelot and A. Ogus. Notes on crystalline cohomology. Princeton University Press, 1978.

• R. Bezrukavnikov, Noncommutative Counterparts of the Springer Resolution (pdf) c.f. §3.1

• R. Bezrukavnikov, I. Mirković, and D. Rumynin, with an appendix by R. Bezrukavnikov and S. Riche. Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2) 167 (2008), no. 3, 945–991.