Derived equivalences for $\hat{I}¦$-Auslander-Yoneda algebras

Abstract

In this paper, we introduce $\hat{I}¦$-Auslander-Yoneda algebras in a triangulated category with $\hat{I}¦$ a parameter set in $\mathbb{N}$, and provide a method to construct new derived equivalences between these $\hat{I}¦$-Auslander-Yoneda algebras (not necessarily Artin algebras), or their quotient algebras, from a given almost $\hat{I}½$-stable derived equivalence. As consequences of our method, we have: (1) Suppose that $A$ and $B$ are representation-finite, self-injective Artin algebras with ${}_{A}X$ and ${}_{B}Y$ additive generators for $A$ and $B$, respectively. If $A$ and $B$ are derived-equivalent, then the $\hat{I}¦$-Auslander-Yoneda algebras of $X$ and $Y$ are derived-equivalent for every admissible set $\hat{I}¦$. 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 $\hat{I}¦$-Auslander-Yoneda algebras of $A\oplus X$ and $A\oplus \hat{I}{\text{c◯}}_A(X)$ are derived-equivalent for every admissible set $\hat{I}¦$, where $\hat{I}\text{c◯}$ is the Heller loop operator. Motivated by these derived equivalences between $\hat{I}¦$-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.

27 pages