On the ring of differential operators of certain regular domains
Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian domain of dimension $d + 1 \geq 2$. Assume $k = A/\mathfrak{m}$ has characteristic zero and that $A$ is not a regular local ring. Let $Sing(A)$ the singular locus of $A$ be defined by an ideal $J$ in $A$. Note $J \neq 0$. Let $ f \in J$ with $f \neq 0$. Set $R = A_f$. Then $R$ is a regular domain of dimension $d$. We show $R$ contains naturally a field $\ell \cong k((X))$. Let $\mathfrak{g}$ be the set of $\ell$-linear derivations of $R$ and let $D(R)$ be the subring of $Hom_\ell(R,R)$ generated by $\mathfrak{g}$ and the multiplication operators defined by elements in the ring $R$. We show that $D(R)$, the ring of $\ell$-linear differential operators on $R$, is a left, right Noetherian ring of global dimension $d$. This enables us to prove Lyubeznik's conjecture on $R$ modulo a conjecture on roots of Bernstein-Sato polynomials over power series rings.