Nakayama automorphisms and modular derivations in filtered deformations

Abstract

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.