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

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 -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.

**1.**Hideto Asashiba,*The derived equivalence classification of representation-finite selfinjective algebras*, J. Algebra**214**(1999), no. 1, 182–221. MR**1684880**, 10.1006/jabr.1998.7706**2.**Michael Barot and Helmut Lenzing,*One-point extensions and derived equivalence*, J. Algebra**264**(2003), no. 1, 1–5. MR**1980681**, 10.1016/S0021-8693(03)00124-8**3.**Dieter Happel,*Triangulated categories in the representation theory of finite-dimensional algebras*, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR**935124****4.**Mitsuo Hoshino and Yoshiaki Kato,*Tilting complexes defined by idempotents*, Comm. Algebra**30**(2002), no. 1, 83–100. MR**1880662**, 10.1081/AGB-120006480**5.**Wei Hu and Changchang Xi,*𝒟-split sequences and derived equivalences*, Adv. Math.**227**(2011), no. 1, 292–318. MR**2782196**, 10.1016/j.aim.2011.01.023**6.**Wei Hu and Changchang Xi,*Derived equivalences and stable equivalences of Morita type, I*, Nagoya Math. J.**200**(2010), 107–152. MR**2747880****7.**Bernhard Keller,*Invariance and localization for cyclic homology of DG algebras*, J. Pure Appl. Algebra**123**(1998), no. 1-3, 223–273. MR**1492902**, 10.1016/S0022-4049(96)00085-0**8.**Yuming Liu and Changchang Xi,*Construction of stable equivalences of Morita type for finite-dimensional algebras. I*, Trans. Amer. Math. Soc.**358**(2006), no. 6, 2537–2560. MR**2204043**, 10.1090/S0002-9947-05-03775-X**9.**Amnon Neeman,*Triangulated categories*, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR**1812507****10.**Jeremy Rickard,*Morita theory for derived categories*, J. London Math. Soc. (2)**39**(1989), no. 3, 436–456. MR**1002456**, 10.1112/jlms/s2-39.3.436**11.**Jeremy Rickard,*Derived equivalences as derived functors*, J. London Math. Soc. (2)**43**(1991), no. 1, 37–48. MR**1099084**, 10.1112/jlms/s2-43.1.37**12.**Jeremy Rickard,*Derived categories and stable equivalence*, J. Pure Appl. Algebra**61**(1989), no. 3, 303–317. MR**1027750**, 10.1016/0022-4049(89)90081-9**13.**J. L. Verdier, Catégories dérivées, etat O,*Lecture Notes in Math.***569**, 262-311, Springer-Verlag, Berlin, 1977.

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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.