Cohomology of tails, Tate–Vogel cohomology, and noncommutative Serre duality over Koszul quiver algebras

The main result of the paper shows that, under Koszul duality between quiver algebras, cohomology of tails is identified with graded Vogel cohomology. As an application, a new proof of the noncommutative Serre duality over generalized Artin–Schelter regular Koszul quiver algebras is given. It is deduced from a similar formula over an arbitrary (i.e., not necessarily Koszul) Frobenius algebra, which turns out to be equivalent to the Auslander–Reiten formula. As another application, it is shown that, over a generalized Artin–Schelter regular Koszul quiver algebra, any algebra automorphism appearing in the noncommutative Serre duality formula is closely related, under Koszul duality, to the Nakayama automorphism of the Koszul-dual algebra.