Derived equivalences for -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

DOI:
https://doi.org/10.1090/S0002-9947-2013-05688-7

Published electronically:
January 9, 2013

MathSciNet review:
3091261

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Abstract | References | Similar Articles | Additional Information

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

Email:
huwei@bnu.edu.cn

**Changchang Xi**

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

Email:
xicc@bnu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-2013-05688-7

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.