# zbMATH — the first resource for mathematics

Skew Calabi-Yau algebras and homological identities. (English) Zbl 1336.16011
Summary: 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 the Nakayama automorphism of a skew Calabi-Yau algebra $$A$$ has trivial homological determinant in case $$A$$ is Noetherian, connected graded, and Koszul.

##### MSC:
 16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) 16S40 Smash products of general Hopf actions 16W50 Graded rings and modules (associative rings and algebras) 16S38 Rings arising from noncommutative algebraic geometry
Full Text: