Suppose that A is a positively filtered algebra such that the associated graded algebra gr A is commutative Calabi-Yau. Then gr A has a canonical Poisson structure with a modular derivation. In general, A is skew Calabi-Yau by a result of Van den Bergh, so A has an invariant, called Nakayama automorphism. A connection between the Nakayama automorphism of A and the modular derivation of gr A is described by using homological determinant as a tool. In particular, it is proved that A is Calabi-Yau if and only if gr A is unimodular as Poisson algebra under some mild assumptions. As an application, the ring of differential operators over a smooth variety is showed to be Calabi-Yau, which was proved by Yekutieli.

假设A是一个正滤波的代数，使得关联的渐变代数gr  A是可交换的Calabi-Yau。则gr  A具有带模推导的规范泊松结构。通常，由于范登·伯格（Van den Bergh）的结果，A会倾斜Calabi-Yau，因此A具有不变性，称为Nakayama自同构。通过使用同源性行列式作为工具来描述A的中山自同构与gr A的模导数 之间的联系。特别地，证明了当且仅当gr A时，A是卡拉比丘 。在某些温和的假设下，它像泊松代数一样是单模的。作为一种应用，在光滑变体上的微分算子环显示为Calabi-Yau，这已得到Yekutieli的证明。