Let $\mathbb{k}$ be an algebraically closed field of positive characteristic, $X$ an affine smooth variety over it. Then the ring of crystalline differential operators on $X$ is generated by $\mathcal{O}(X)$ and $\operatorname{Der}\_\mathbb{k} \mathcal{O}(X)$ with relations $f.\partial=f\partial$, $\partial.f-f.\partial=\partial(f)$, $\partial.\partial'-\partial'.\partial=[\partial,\partial']$, $f \in \mathcal{O}(X), \partial,\partial' \in \operatorname{Der}\_\mathbb{k} \mathcal{O}(X)$. An important example are the Weyl algebras over $\mathbb{k}$.


I'm looking for papers that discuss ring theoretical aspects of this ring. I know that it is a left and right Noetherian domain, finitely generated, and the description of its center, but nothing more.


I am also interested in the question of when this notion of differential operators is better suited than the classical Grothendieck's notion of differential operator.



Reference request on rings of crystalline differential operators
Let $\pi: T^\ast X \to X$ be the projection of the cotangent bundle and $-'$ denote Frobenius twist.
Then $D(X)$ is an Azumaya algebra over its center $\pi\_\ast \mathcal O\_{T^\*X'}$. This appears in Roman Bezrukavnikov, Ivan Mirković, Dmitriy Rumynin: "Localization of modules for a semisimple Lie algebra in prime characteristic."


