  We show that many noetherian Hopf algebras A have a rigid dualising complex R
with R isomorphic to ^{\nu}A^1 [d]. Here, d is the injective dimension of the
algebra and \nu is 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 \nu the Nakayama automorphism of A. We prove that \nu =
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 \nu for 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.
