Algebras stably equivalent to trivial extensions of hereditary algebras of type à n

Zygmunt Pogorzały

Colloquium Mathematicae (1993)

  • Volume: 66, Issue: 2, page 265-281
  • ISSN: 0010-1354

Abstract

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The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting equivalent algebras then their trivial extensions T(A) and T(B) are stably equivalent. Consequently, there is a special need to describe algebras that are stably equivalent to the trivial extensions of tame hereditary algebras. In the paper, there are studied algebras which are stably equivalent to the trivial extensions of hereditary algebras of type à n , that is, algebras given by quivers whose underlying graphs are of type à n . These algebras are isomorphic to the trivial extensions of very nice algebras (see Theorem 1). Moreover, in view of [1, 8], Theorem 2 shows that every stable equivalence of such algebras is induced in some sense by a derived equivalence of well chosen subalgebras. In our study of stable equivalence, we shall use methods and results from [11]. We shall also use freely information on Auslander-Reiten sequences which can be found in [2].

How to cite

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Pogorzały, Zygmunt. "Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$." Colloquium Mathematicae 66.2 (1993): 265-281. <http://eudml.org/doc/210248>.

@article{Pogorzały1993,
abstract = {The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting equivalent algebras then their trivial extensions T(A) and T(B) are stably equivalent. Consequently, there is a special need to describe algebras that are stably equivalent to the trivial extensions of tame hereditary algebras. In the paper, there are studied algebras which are stably equivalent to the trivial extensions of hereditary algebras of type $Ã_n$, that is, algebras given by quivers whose underlying graphs are of type $Ã_n$. These algebras are isomorphic to the trivial extensions of very nice algebras (see Theorem 1). Moreover, in view of [1, 8], Theorem 2 shows that every stable equivalence of such algebras is induced in some sense by a derived equivalence of well chosen subalgebras. In our study of stable equivalence, we shall use methods and results from [11]. We shall also use freely information on Auslander-Reiten sequences which can be found in [2].},
author = {Pogorzały, Zygmunt},
journal = {Colloquium Mathematicae},
keywords = {biserial algebras; basic connected hereditary algebras; trivial extensions; minimal injective cogenerators; stable module categories; finitely generated right modules; tilting-cotilting equivalent to hereditary algebras},
language = {eng},
number = {2},
pages = {265-281},
title = {Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$},
url = {http://eudml.org/doc/210248},
volume = {66},
year = {1993},
}

TY - JOUR
AU - Pogorzały, Zygmunt
TI - Algebras stably equivalent to trivial extensions of hereditary algebras of type $Ã_n$
JO - Colloquium Mathematicae
PY - 1993
VL - 66
IS - 2
SP - 265
EP - 281
AB - The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting equivalent algebras then their trivial extensions T(A) and T(B) are stably equivalent. Consequently, there is a special need to describe algebras that are stably equivalent to the trivial extensions of tame hereditary algebras. In the paper, there are studied algebras which are stably equivalent to the trivial extensions of hereditary algebras of type $Ã_n$, that is, algebras given by quivers whose underlying graphs are of type $Ã_n$. These algebras are isomorphic to the trivial extensions of very nice algebras (see Theorem 1). Moreover, in view of [1, 8], Theorem 2 shows that every stable equivalence of such algebras is induced in some sense by a derived equivalence of well chosen subalgebras. In our study of stable equivalence, we shall use methods and results from [11]. We shall also use freely information on Auslander-Reiten sequences which can be found in [2].
LA - eng
KW - biserial algebras; basic connected hereditary algebras; trivial extensions; minimal injective cogenerators; stable module categories; finitely generated right modules; tilting-cotilting equivalent to hereditary algebras
UR - http://eudml.org/doc/210248
ER -

References

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