Derived equivalences for AuslanderYoneda algebras
Authors:
Wei Hu and Changchang Xi
Journal:
Trans. Amer. Math. Soc. 365 (2013), 56815711
MSC (2010):
Primary 18E30, 16G10; Secondary 18G15, 16L60
Published electronically:
January 9, 2013
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Abstract: In this paper, we first define a new family of Yoneda algebras, called AuslanderYoneda 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 AuslanderYoneda 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 selfinjective Artin algebra, then, for any module and for any admissible set in , the AuslanderYoneda algebras of and are derived equivalent, where is the Heller loop operator. (2) Suppose that and are representationfinite selfinjective algebras with additive generators and , respectively. If and are derived equivalent, then so are the AuslanderYoneda 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 AuslanderYoneda algebras, we show, among other results, that a derived equivalence between two basic selfinjective 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:
http://dx.doi.org/10.1090/S000299472013056887
PII:
S 00029947(2013)056887
Keywords:
AuslanderYoneda 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.
