A new invariant of stable equivalences of Morita type
HTML articles powered by AMS MathViewer
- by Zygmunt Pogorzały PDF
- Proc. Amer. Math. Soc. 131 (2003), 343-349 Request permission
Abstract:
It was proved in an earlier paper by the author that the Hochschild cohomology algebras of self-injective algebras are invariant under stable equivalences of Morita type. In this note we show that the orbit algebra of a self-injective algebra $A$ (considered as an $A$-$A$-bimodule) is also invariant under stable equivalences of Morita type, where the orbit algebra is the algebra of all stable $A$-$A$-bimodule morphisms from the non-negative Auslander-Reiten translations of $A$ to $A$.References
- Hideto Asashiba, A covering technique for derived equivalence, J. Algebra 191 (1997), no. 1, 382–415. MR 1444505, DOI 10.1006/jabr.1997.6906
- Hideto Asashiba, The derived equivalence classification of representation-finite selfinjective algebras, J. Algebra 214 (1999), no. 1, 182–221. MR 1684880, DOI 10.1006/jabr.1998.7706
- Shi Ping Liu, On short cycles in a module category, J. London Math. Soc. (2) 51 (1995), no. 1, 62–74. MR 1310722, DOI 10.1112/jlms/51.1.62
- Michel Broué, Equivalences of blocks of group algebras, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 1–26. MR 1308978, DOI 10.1007/978-94-017-1556-0_{1}
- Peter Gabriel, Auslander-Reiten sequences and representation-finite algebras, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, Springer, Berlin, 1980, pp. 1–71. MR 607140
- 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, DOI 10.1017/CBO9780511629228
- Dieter Happel, Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108–126. MR 1035222, DOI 10.1007/BFb0084073
- Alex Heller, The loop-space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960), 382–394. MR 116045, DOI 10.1090/S0002-9947-1960-0116045-4
- Otto Kerner, Minimal approximations, orbital elementary modules, and orbit algebras of regular modules, J. Algebra 217 (1999), no. 2, 528–554. MR 1700513, DOI 10.1006/jabr.1998.7815
- H. Lenzing, Wild canonical algebras and rings of automorphic forms, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 191–212. MR 1308987
- Markus Linckelmann, Stable equivalences of Morita type for self-injective algebras and $p$-groups, Math. Z. 223 (1996), no. 1, 87–100. MR 1408864, DOI 10.1007/PL00004556
- Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
- Z. Pogorzały, Invariance of Hochschild cohomology algebras under stable equivalences of Morita type, J. Math. Soc. Japan 53(2001), 913–918.
- Z. Pogorzały, Left-right projective bimodules and stable equivalences of Morita type, Colloq. Math. Vol.88 (2) (2001), 243-255.
- Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. MR 1099084, DOI 10.1112/jlms/s2-43.1.37
Additional Information
- Zygmunt Pogorzały
- Affiliation: Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
- Email: zypo@mat.uni.torun.pl
- Received by editor(s): May 2, 2001
- Received by editor(s) in revised form: September 6, 2001
- Published electronically: June 5, 2002
- Communicated by: Martin Lorenz
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 343-349
- MSC (2000): Primary 16D50; Secondary 16G20
- DOI: https://doi.org/10.1090/S0002-9939-02-06553-X
- MathSciNet review: 1933322
Dedicated: Dedicated to Professor Idun Reiten on the occasion of her sixtieth birthday