Some algebras with trivial rings of differential operators
Abstract
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 nonsingular 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 pushforward $F_*^eR$ 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 $D_k(R)\to D_k(R, k)$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.09184
 arXiv:
 arXiv:2404.09184
 Bibcode:
 2024arXiv240409184M
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 13A35;
 13N10;
 13N15;
 16S32
 EPrint:
 Comments welcome