Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras

Abstract

We show that many noetherian Hopf algebras A have a rigid dualising complex R with R isomorphic to ^νA^1 [d]. Here, d is the injective dimension of the algebra and νis a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call νthe Nakayama automorphism of A. We prove that ν= S^2\XXi, where S is the antipode of A and \XXi is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non-zero in the top dimension d, when A is an Artin-Schelter regular noetherian Hopf algebra of global dimension d. (Twisted) Poincare duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating νfor A using also the opposite coalgebra structure, we determine a formula for S^4 generalising a 1976 formula of Radford for A finite dimensional. Applications of the results to the cases where A is PI, an enveloping algebra, a quantum group, a quantised function algebra and a group algebra are outlined.