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 $\Der_F (R)$ is a finitely generated projective $R$-module of rank $n$ such that $R_m\otimes_R \Der_F (R)=\Der_F(R_m)$. We show that the associated graded ring of the ring of differentiable operators, $D(R,F)$, is a commutative Noetherian regular with unity and pure graded dimension equal to $2\dim(R)$. 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, $H^i_I(R)$, 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 $p$, $S_p$, is such a ring. As a consequence, the set of associated primes of $H^i_I (S)$ that does not contain $p$ is finite. Moreover, we prove this finiteness property for a larger class of functors.