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Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components

Published online by Cambridge University Press:  18 May 2009

Zygmunt Pogorzały
Zygmunt Pogorzały
Affiliation:
Faculty of Mathematics and Informatics Nicholas Copernicus Universityul. Chopina 12/18, 87–100 Toruń, Poland e-mail [email protected]
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Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 1 , March 1998 , pp. 1 - 19
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

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