Derived equivalences for $Φ$-Auslander-Yoneda algebras

In this paper, we introduce $Φ$-Auslander-Yoneda algebras in a triangulated category with $Φ$ a parameter set in $\mathbb N$, and provide a method to construct new derived equivalences between these $Φ$-Auslander-Yoneda algebras (not necessarily Artin algebras), or their quotient algebras, from a given almost $ν$-stable derived equivalence. As consequences of our method, we have: (1) Suppose that $A$ and $B$ are representation-finite, self-injective Artin algebras with $_AX$ and $_BY$ additive generators for $A$ and $B$, respectively. If $A$ and $B$ are derived-equivalent, then the $Φ$-Auslander-Yoneda algebras of $X$ and $Y$ are derived-equivalent for every admissible set $Φ$. In particular, the Auslander algebras of $A$ and $B$ are both derived-equivalent and stably equivalent. (2) For a self-injective Artin algeba $A$ and an $A$-module $X$, the $Φ$-Auslander-Yoneda algebras of $A\oplus X$ and $A\oplus Ω_A(X)$ are derived-equivalent for every admissible set $Φ$, where $Ω$ is the Heller loop operator. Motivated by these derived equivalences between $Φ$-Auslander-Yoneda algebras, we consider constructions of derived equivalences for quotient algebras, and show, among others, that a derived equivalence between two basic self-injective algebras may transfer to a derived equivalence between their quotient algebras obtained by factorizing out socles.

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