Differential operators, retracts, and toric face rings

TL;DR

This paper provides explicit descriptions of differential operators for toric face rings and related structures, introduces a criterion for Gorenstein property, and develops a retract-based technique applicable in characteristic zero.

Contribution

It introduces a new retract-based method to describe differential operators on toric face rings and related quotients, extending understanding in characteristic zero.

Findings

Explicit descriptions of differential operators for toric face rings.

Identification of differential operators induced from ambient rings.

A criterion for the Gorenstein property based on differential operators.

Abstract

We give explicit descriptions of rings of differential operators of toric face rings in characteristic $0$. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators are induced by differential operators on the ambient ring. Lastly, we provide a criterion for the Gorenstein property of a normal affine semigroup ring in terms of its differential operators. Our main technique is to realize the k-algebras we study in terms of a suitable family of their algebra retracts in a way that is compatible with the characterization of differential operators. This strategy allows us to describe differential operators of any k-algebra realized by retracts in terms of the differential operators on these retracts, without restriction on char(k).