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 $A♯H$ 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.