  A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which
allows for a non-trivial Nakayama automorphism. We prove three homological
identities about the Nakayama automorphism and give several applications. The
identities we prove show (i) how the Nakayama automorphism of a smash product
algebra A # H is related to the Nakayama automorphisms of a graded skew
Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it;
(ii) how the Nakayama automorphism of a graded twist of A is related to the
Nakayama automorphism of A; and (iii) that Nakayama automorphism of a skew
Calabi-Yau algebra A has trivial homological determinant in case A is
noetherian, connected graded, and Koszul.
