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

摘要

Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) withcoefficients in a domain k, and K its associated simplicial complex. Also letD_k(R) be the ring of k-linear differential operators on R. We give twodifferent descriptions of the two-sided ideal structure of D_k(R) as being inbijection with certain well-known subcomplexes of K; one based on explicitcomputation in the Weyl algebra, valid in any characteristic, and one valid incharacteristic 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 anddifferent interpretation using these techniques.