Smash Products of Calabi-Yau Algebras by Hopf Algebras

Patrick Le Meur

Published 2015-12-03Version 1

Given a Hopf algebra H and an H-module differential graded (dg) algebra A, this text investigates the smash product A#H from the viewpoint of Calabi-Yau duality. First it proves that all Hopf algebras with Van den Bergh duality have invertible antipode. Next it describes the inverse dualising object of A#H when both A and H are homologically smooth and the antipode of H is invertible. This is applied to the standard constructions of Calabi-Yau dg algebras. Finally, this description is applied to prove that if A is an algebra and both A and H have Van den Bergh duality (or, are skew-Calabi-Yau) then A#H has Van den Bergh duality (or, is skew-Calabi-Yau, respectively). In the case of skew-Calabi-Yau algebras, a Nakayama automorphism for A#H is expressed in terms of those of A and H and using the weak homological determinant of the action of H on A; The weak homological determinant is introduced to replace the homological determinant when this latter notion may not be defined. This yields some sufficient conditions for A#H to be Calabi-Yau in the more particular case where A is connected graded. The results in this text extend results proved previously by several authors and that use technical assumptions (on A or H) that appear to be unnecessary.