  Let H be a Hopf algebra and A be an H-module algebra. This article
investigates when the smash product A#H is (skew) Calabi-Yau, has Van den Bergh
duality or is Artin-Schelter regular or Gorenstein. In particular, if A and H
are skew Calabi-Yau, then so is A#H and its Nakayama automorphism is expressed
using the ones of A and H. This is based on a description of the inverse
dualising complex of A#H when A is a homologically smooth dg algebra and H is
homologically smooth and with invertible antipode. This description is also
used to explain the compatibility of standard constructions of Calabi-Yau dg
algebras with taking smash products.
