2009/12/03 by Wei Hu, Hu, Wei, Changchang Xi +1
Mathematics · #16G10 #16S50 #18E30 #18G15 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16G10 #msc:16S50 #msc:18E30 #msc:18G15
paper · pdf · doi:10.48550/arxiv.0912.0647
27 pages
arxiv created 2010/02/23 · arxiv updated 2010/02/26
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⊕ X and A⊕ ΩA(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.