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 {\it 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 {\it regular} local rings $R$ (with fraction field a function field) whose Frobenius push-forward $F_*^eR$ is {\it 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 $D_k(R)\to D_k(R, k)$.