On the ring of differential operators of certain regular domains

Abstract:

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 $\text {Sing}(A)$, the singular locus of $A$, be defined by an ideal $J$ in $A$. Note that $J \neq 0$. Let $f \in J$ with $f \neq 0$. Set $R = A_f$. Then $R$ is a regular domain of dimension $d$. We show that $R$ contains naturally a field $\mathbb {L} \cong k((X))$ such that $D(R)$, the ring of $\mathbb {L}$-linear differential operators on $R$, is a left and right Noetherian ring of global dimension $d$. This enables us to prove Lyubeznik’s conjecture on $R$ regarding finiteness of associate primes of local cohomology modules of $R$.