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 $\mathrm{char}<!--FUNCTION\; APPLICATION-->(k)$.

我们在特征上给出了复曲面环微分算子环的显式描述$0$。对于根式单项式理想的正规仿射半群环的商，我们还确定了它们的哪些微分算子是由环境环上的微分算子导出的。最后，我们根据微分算子提供了正规仿射半群环的 Gorenstein 性质的判据。

我们的主要技术是以与微分算子的表征兼容的方式，以合适的代数族来实现我们研究的 k 代数。这种策略允许我们用这些缩回上的微分算子来描述由缩回实现的任何 k 代数的微分算子，而不受限制$\mathrm{字符}<!--FUNCTION\; APPLICATION-->（k）$。