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

Authors: Wei Hu and Changchang Xi
Journal: Trans. Amer. Math. Soc. 365 (2013), 5681-5711
MSC (2010): Primary 18E30, 16G10; Secondary 18G15, 16L60
Published electronically: January 9, 2013
MathSciNet review: 3091261
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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.

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Additional Information

Wei Hu
Affiliation: School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, 100875 Beijing, People’s Republic of China

Changchang Xi
Affiliation: School of Mathematical Sciences, Capital Normal University, 100048 Beijing, People’s Republic of China

Keywords: Auslander-Yoneda algebra, derived equivalence, quotient algebra, tilting complex
Received by editor(s): November 18, 2010
Received by editor(s) in revised form: August 3, 2011
Published electronically: January 9, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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