2024/04/14 by Alapan Mukhopadhyay, Mukhopadhyay, Alapan, Karen E. Smith +1
Computer Science · Mathematics · #13A35 #13N10 #13N15 #16S32 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2404.09184
openalex publication_date 2024/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \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∈ \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 Dk(R)→ Dk(R, k).