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2014 | 12 | 6 | 813-823
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Orbit algebras that are invariant under stable equivalences of Morita type

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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.
Twórcy
Bibliografia
  • [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
  • [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
  • [3] Bergh P.A., Orbit algebras and periodicity, Colloq. Math., 2009, 114(2), 245–252 http://dx.doi.org/10.4064/cm114-2-7
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
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bwmeta1.element.doi-10_2478_s11533-013-0385-7
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