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 Dₖ(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_*ᵉR is { indecomposable} as an R-module for all e∈ 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 N-graded case: for a Noetherian local k-algebra $(R, m, k)$, D-simplicity of R is equivalent to surjectivity of the natural map Dₖ(R)→ Dₖ(R, k).