Derived equivalences for $\Phi$-Auslander-Yoneda algebras

Abstract:

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