In this note we show that there are a lot of orbit algebras that are invariant under stable equivalences of Morita type between self-injective algebras. There are also indicated some links between Auslander-Reiten periodicity of bimodules and noetherianity of their orbit algebras.
@article{bwmeta1.element.doi-10_2478_s11533-013-0385-7,
author = {Zygmunt Pogorza\l y},
title = {Orbit algebras that are invariant under stable equivalences of Morita type},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {813-823},
zbl = {1331.16007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0385-7}
}
Zygmunt Pogorzały. Orbit algebras that are invariant under stable equivalences of Morita type. Open Mathematics, Tome 12 (2014) pp. 813-823. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0385-7/
[1] Assem I., Simson D., Skowronski A., Elements of the Representation Theory of Associative Algebras, I, London Math. Soc. Stud. Texts, 65, Cambridge University Press, Cambridge, 2006 http://dx.doi.org/10.1017/CBO9780511614309 | Zbl 1092.16001
[2] Auslander M., Reiten I., Representation theory of Artin algebras III. Almost split sequences, Comm. Algebra, 1975, 3(3), 239–294 http://dx.doi.org/10.1080/00927877508822046 | Zbl 0331.16027
[3] Bergh P.A., Orbit algebras and periodicity, Colloq. Math., 2009, 114(2), 245–252 http://dx.doi.org/10.4064/cm114-2-7 | Zbl 1187.16014
[4] Brenner S., Butler M.C.R., King A.D., Periodic algebras which are almost Koszul, Algebr. Represent. Theory, 2002, 5(4), 331–367 http://dx.doi.org/10.1023/A:1020146502185 | Zbl 1056.16003
[5] Broué M., Equivalences of blocks of group algebras, In: Finite-Dimensional Algebras and Related Topics, Ottawa, August 10–18, 1992, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer, Dordrecht, 1994, 1–26
[6] Dugas A., Periodic resolutions and self-injective algebras of finite type, J. Pure Appl. Algebra, 2010, 214(6), 990–1000 http://dx.doi.org/10.1016/j.jpaa.2009.09.012 | Zbl 1214.16006
[7] Green E.L., Snashall N., Solberg Ø., The Hochschild cohomology ring of a selfinjective algebra of finite representation type, Proc. Amer. Math. Soc., 2003, 131(11), 3387–3393 http://dx.doi.org/10.1090/S0002-9939-03-06912-0 | Zbl 1061.16017
[8] Happel D., Hochschild cohomology of finite-dimensional algebras, In: Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année, Paris, 1987/1988, Lecture Notes in Math., 1404, Springer, Berlin, 1989, 108–126 http://dx.doi.org/10.1007/BFb0084073
[9] Heller A., The loop-space functor in homological algebra, Trans. Amer. Math. Soc., 1960, 96, 382–394 http://dx.doi.org/10.1090/S0002-9947-1960-0116045-4 | Zbl 0096.25502
[10] Kerner O., Minimal approximations, orbital elementary modules, and orbit algebras of regular modules, J. Algebra, 1999, 217(2), 528–554 http://dx.doi.org/10.1006/jabr.1998.7815
[11] Lenzing H., Wild canonical algebras and rings of automorphic forms, In: Finite-Dimensional Algebras and Related Topics, Ottawa, August 10–18, 1992, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer, Dordrecht, 1994, 191–212 | Zbl 0895.16004
[12] Linckelmann M., Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z., 1996, 223(1), 87–100 http://dx.doi.org/10.1007/BF02621590 | Zbl 0866.16004
[13] Pogorzały Z., Left-right projective bimodules and stable equivalences of Morita type, Colloq. Math., 2001, 88(2), 243–255 http://dx.doi.org/10.4064/cm88-2-6 | Zbl 0985.16004
[14] Pogorzały Z., Invariance of Hochschild cohomology algebras under stable equivalences of Morita type, J. Math. Soc. Japan, 2001, 53(4), 913–918 http://dx.doi.org/10.2969/jmsj/05340913 | Zbl 1043.16008
[15] Pogorzały Z., A new invariant of stable equivalences of Morita type, Proc. Amer. Math. Soc., 2002, 131(2), 343–349 http://dx.doi.org/10.1090/S0002-9939-02-06553-X | Zbl 1062.16003
[16] Pogorzały Z., On Galois coverings of the enveloping algebras of self-injective Nakayama algebras, Comm. Algebra, 2003, 31(6), 2985–2999 http://dx.doi.org/10.1081/AGB-120021904 | Zbl 1050.16011
[17] Pogorzały Z., On the Auslander-Reiten periodicity of self-injective algebras, Bull. London Math. Soc., 2004, 36(2), 156–168 http://dx.doi.org/10.1112/S0024609303002716 | Zbl 1067.16031
[18] Rickard J., Derived equivalences as derived functors, J. London Math. Soc., 1991, 43(1), 37–48 http://dx.doi.org/10.1112/jlms/s2-43.1.37 | Zbl 0683.16030
[19] Schulz R., Boundedness and periodicity of modules over QF rings, J. Algebra, 1986, 101(2), 450–469 http://dx.doi.org/10.1016/0021-8693(86)90204-8
[20] Snashall N., Solberg Ø., Support varieties and Hochschild cohomology rings, Proc. London Math. Soc., 2004, 88(3), 705–732 http://dx.doi.org/10.1112/S002461150301459X | Zbl 1067.16010