On some Rings of differentiable type

Let K be a field of characteristic 0 and S=Kx₁, , xₘ/I be an affine domain. Consider R=SP where P Spec (S) such that R is regular. In this paper we construct a field F which is contained in R such that (1) The residue field of R is a finite extension of F. (2) DF (R), the ring of F-linear differential operators on R is left and right Noetherian with finite global dimension. (3) The Bernstein class of DF (R) is closed under localization at one element of R. We also prove a similar result for Rʰ, the Henselization of R. As an application we prove that DF (R) DF (R) P E ( (P) ) where E ( (P) ) is the injective hull of the residue field of R.