Derived equivalences for Φ-Auslander-Yoneda algebras

Abstract

In this paper, we first define a new family of Yoneda algebras, called Φ \Phi -Auslander-Yoneda algebras, in triangulated categories by introducing the notion of admissible sets Φ \Phi in N \mathbb N , which includes higher cohomologies indexed by Φ \Phi , and then present a general method to construct a family of new derived equivalences for these Φ \Phi -Auslander-Yoneda algebras (not necessarily Artin algebras), where the choices of the parameters Φ \Phi are rather abundant. Among applications of our method are the following results: (1) if A A is a self-injective Artin algebra, then, for any A A -module X X and for any admissible set Φ \Phi in N \mathbb N , the Φ \Phi -Auslander-Yoneda algebras of A ⊕ X A\oplus X and A ⊕ Ω A ( X ) A\oplus \Omega _A(X) are derived equivalent, where Ω \Omega is the Heller loop operator. (2) Suppose that A A and B B are representation-finite self-injective algebras with additive generators A X _AX and B Y _BY , respectively. If A A and B B are derived equivalent, then so are the Φ \Phi -Auslander-Yoneda algebras of X X and Y Y for any admissible set Φ \Phi . In particular, the Auslander algebras of A A and B B are derived equivalent. The converse of this statement is open. Further, motivated by these derived equivalences between Φ \Phi -Auslander-Yoneda algebras, we show, among other results, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factoring out socles.