Smash products of Calabi–Yau algebras by Hopf algebras

Abstract

Let H be a Hopf algebra and A be an H -module algebra. This article investigates when the smash product A\sharp 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\sharp 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\sharp 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.