Sheaf of differential operators

Definition

Let $X$ be a smooth algebraic variety over a field of characteristic $0$. The sheaf of differential operators $\mathcal D_X$ is the sheaf of associative algebras generated locally by $\mathcal O_X$ and the tangent sheaf $\mathcal T_X$. Its relations include multiplication in $\mathcal O_X$, the $\mathcal O_X$-module structure on $\mathcal T_X$, and

for local functions $f$ and vector fields $\xi,\eta$. Equivalently, $\mathcal D_X$ is the universal enveloping algebra of the Lie algebroid $\mathcal T_X$.

It has the order filtration $\mathcal D_X^{\leq0}\subseteq\mathcal D_X^{\leq1}\subseteq\cdots$, and the associated graded algebra is canonically $\operatorname{Sym}_{\mathcal O_X}\mathcal T_X$.