Two-sided ideals in the ring of differential operators on a Stanley-Reisner ring

TL;DR

This paper characterizes the two-sided ideals of the differential operator ring on a Stanley-Reisner ring, linking them to subcomplexes of the associated simplicial complex, and provides new proofs and descriptions in different characteristics.

Contribution

It offers two novel descriptions of the two-sided ideal structure of D_k(R), connecting algebraic and combinatorial aspects, and presents a new proof of Traves' module structure result.

Findings

Two descriptions of two-sided ideals in D_k(R) linked to subcomplexes of K

Explicit computation in Weyl algebra valid in any characteristic

Frobenius splitting approach valid in characteristic p

Abstract

Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) with coefficients in a domain k, and K its associated simplicial complex. Also let D_k(R) be the ring of k-linear differential operators on R. We give two different descriptions of the two-sided ideal structure of D_k(R) as being in bijection with certain well-known subcomplexes of K; one based on explicit computation in the Weyl algebra, valid in any characteristic, and one valid in characteristic p based on the Frobenius splitting of R. A result of Traves [Tra99] on the D_k(R)-module structure of R is also given a new proof and different interpretation using these techniques.