On some Rings of differentiable type

Let K be a field of characteristic 0 and S=K[x₁,…,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(κ(P)) where E(κ(P)) is the injective hull of the residue field of R.