D-modules with finite support are semi-simple

TL;DR

This paper provides an explicit decomposition of certain D-modules with finite support over regular local algebras into simple modules, using Pochhammer differential operators, revealing their semi-simple structure.

Contribution

It introduces a concrete method to decompose D-modules with finite support into simple components, expanding understanding of their semi-simplicity in algebraic geometry.

Findings

Explicit decomposition of D_R/D_R m_R^{n+1} into simple modules

Use of Pochhammer differential operators for generators

Demonstration that these D-modules are semi-simple

Abstract

Let $(R, \mf, k_R)$ be regular local $k$-algebra satisfying the weak Jacobian criterion, such that $k_R/k$ is an algebraic field extension. Let $D_R$ be the ring of $k$-linear differential operators of $R$. We give an explicit decomposition of the $D_R$-module $D_R/D_R \mf_R^{n+1}$ as a direct sum of simple modules, all isomorphic to $D_R/D_R \mf$, where certain "Pochhammer" differential operators are used to describe generators of the simple components.