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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
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