Some Algebras With Trivial Rings of Differential Operators

Let $k$ be an arbitrary field. We construct examples of regular local $k$-algebras $R$ (of positive dimension) for which the ring of differential operators $D_{k}(R)$ is trivial in the sense that it contains no operators of positive order. The examples are excellent in characteristic zero but not in positive characteristic. These rings can be viewed as being non-singular but they are not simple as $D$-modules, laying to rest speculation that $D$-simplicity might characterize a nice class of singularities in general. In prime characteristic, the construction also provides examples of regular local rings $R$ (with fraction field a function field) whose Frobenius push-forward $F_{*}^{e}R$ is indecomposable as an $R$-module for all $e\in \mathbb N$. Along the way, we investigate hypotheses on a local ring $(R, m)$ under which $D$-simplicity for $R$ is equivalent to $D$-simplicity for its $m$-adic completion, and give examples of rings for which the differential operators do not behave well under completion. We also generalize a characterization of $D$-simplicity due to Jeffries in the $\mathbb N$-graded case: for a Noetherian local $k$-algebra $(R, m, k)$, $D$-simplicity of $R$ is equivalent to surjectivity of the natural map $D_{k}(R)\to D_{k}(R, k)$.