On certain rings of differentiable type and finiteness properties of local cohomology

Let R be a commutative F-algebra, where F is a field of characteristic 0, satisfying the following conditions: R is equidimensional of dimension n, every residual field with respect to a maximal ideal is an algebraic extension of F, and DerF(R) is a finitely generated projective R-module of rank n such that Rm⊗RDerF(R)=DerF(Rm). We show that the associated graded ring of the ring of differential operators, D(R,F), is a commutative Noetherian regular with unity and pure graded dimension equal to 2dim(R). Moreover, we prove that D(R,F) has weak global dimension equal to dim(R) and that its Bernstein class is closed under localization at one element. Using these properties of D(R,F), we show that the set of associated primes of every local cohomology module, , is finite. If (S,m,K) is a complete regular local ring of mixed characteristic p>0, we show that the localization of S at p, Sp, is such a ring. As a consequence, the set of associated primes of that does not contain p is finite. Moreover, we prove this finiteness property for a larger class of functors.