  According to a statement by Pavel Etingof, in the special case of an affine
variety $X$ with a faithful action by a finite group $G$, the sheaf of
(twisted) Cherednik algebras $\mathcal{H}_{1, c, \psi, X, G}$ with formal
parameters $c, \psi$ is a universal formal deformation of $\mathcal{D}_X\rtimes
G$ where $\mathcal{D}_X$ is the sheaf of differential operators on $X$. In the
current note, we generalize Etingof's result to the non-affine case. We prove
that for a generic smooth analytic or algebraic variety $X$, the sheaf
$\mathcal{H}_{1, c, \psi, X, G}$ with formal $c$ and $\psi$ is a universal
filtered formal deformation of $\mathcal{D}_X\rtimes G$. To that aim, we first
construct quasi-isomorphisms between the Hochschild (co)chain complex of
$\mathcal{D}_X\rtimes G$ and the $G$-invariant part of the direct sum over all
elements $g$ in $G$ of sheaves of holomorphic differential forms on the
cotangent bundles of the $g$-fixed point submanifolds in $X$. Finally, we
combine these quasi-isomorphisms with results from the theory of algebraic
extensions for sheaves of filtered associative algebras to establish a
bijective correspondence between the space of isomorphism classes of filtered
infinitesimal deformations of $\mathcal{D}_X\rtimes G$ and the parameter space
of $\mathcal{H}_{1, c, \psi, X, G}$.
