On the ring of differential operators of certain regular domains

TL;DR

This paper investigates the structure of differential operator rings on certain regular domains derived from singular Noetherian domains, proving their Noetherian property and global dimension, and relating to Lyubeznik's conjecture.

Contribution

It establishes that the ring of differential operators on these regular domains is Noetherian with finite global dimension, advancing understanding of their algebraic properties.

Findings

The differential operator ring is left and right Noetherian.

The global dimension of the differential operator ring is equal to the dimension of the domain.

Supports Lyubeznik's conjecture under certain conditions.

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 $Sing(A)$ 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.