A new invariant of stable equivalences of Morita type

Author:
Zygmunt Pogorzaly

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

Published electronically:
June 5, 2002

MathSciNet review:
1933322

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 (considered as an --bimodule) is also invariant under stable equivalences of Morita type, where the orbit algebra is the algebra of all stable --bimodule morphisms from the non-negative Auslander-Reiten translations of to .

**[1]**H. Asashiba,*A covering technique for derived equivalence*, J. Algebra 191(1997), 382-415. MR**98b:16009****[2]**H. Asashiba,*The derived equivalence classification of representation-finite selfinjective algebras*, J. Algebra 214(1999), 182-221. MR**2000g:16019****[3]**M. Auslander, I. Reiten and S. Smalo,*Representation Theory of Artin Algebras*, Cambridge Studies in Advanced Math. Vol. 36, Cambridge Univ. Press (Cambridge, 1995). MR**96e:16015****[4]**M. Broué,*Equivalences of Blocks of Group Algebras*, in: V. Dlab and L.L. Scott (eds.) Finite Dimensional Algebras and Related Topics, NATO ASI Series C Vol. 424, Kluwer Academic Press (Dodrecht, 1992), 1-26. MR**97c:20004****[5]**P. Gabriel,*Auslander-Reiten sequences and representation-finite algebras*, Lecture Notes in Math. 831(Springer-Verlag, Berlin, 1980), 1-71. MR**82i:16030****[6]**D. Happel,*Triangulated categories in the representation theory of finite-dimensional algebras*, London Math. Soc. Lecture Notes 119, Cambridge Univ. Press (Cambridge, 1988). MR**89e:16035****[7]**D. Happel,*Hochschild cohomology of finite-dimensional algebras*, in: Seminair d'Algebre P. Dubriel et M-P.Maliavin, Lecture Notes in Math. 1404(Springer-Verlag, Berlin, 1989), 108-126. MR**91b:16012****[8]**A. Heller,*The loop-space functor in homological algebra*, Trans. Amer. Math. Soc. 96(1960), 382-394. MR**22:6840****[9]**O. Kerner,*Minimal approximations, orbital elementary modules, and orbit algebras of regular modules*, J. Algebra 217(1999), 528-554. MR**2000e:16018****[10]**H. Lenzing,*Wild Canonical Alebras and Rings of Automorphic Forms*, in: V. Dlab and L.L. Scott (eds.) Finite Dimensional Algebras and Related Topics, NATO ASI Series C Vol. 424, Kluwer Academic Press (Dodrecht, 1992), 191-212. MR**95m:16008****[11]**M. Linckelmann,*Stable equivalences of Morita type for self-injective algebras and**-groups*, Math. Z. 223(1996), 87-100. MR**97j:20011****[12]**S. Mac Lane,*Homology*, (Springer-Verlag, Berlin, 1963). MR**28:122****[13]**Z. Pogorzay,*Invariance of Hochschild cohomology algebras under stable equivalences of Morita type*, J. Math. Soc. Japan 53(2001), 913-918.**[14]**Z. Pogorzay,*Left-right projective bimodules and stable equivalences of Morita type*, Colloq. Math. Vol.88 (2) (2001), 243-255.**[15]**J. Rickard,*Derived equivalences as derived functors*, J. London Math. Soc. (2) 43(1991), 37-48. MR**92b:16043**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16D50,
16G20

Retrieve articles in all journals with MSC (2000): 16D50, 16G20

Additional Information

**Zygmunt Pogorzaly**

Affiliation:
Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Email:
zypo@mat.uni.torun.pl

DOI:
https://doi.org/10.1090/S0002-9939-02-06553-X

Received by editor(s):
May 2, 2001

Received by editor(s) in revised form:
September 6, 2001

Published electronically:
June 5, 2002

Dedicated:
Dedicated to Professor Idun Reiten on the occasion of her sixtieth birthday

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2002
American Mathematical Society