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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Tilted algebras


Authors: Dieter Happel and Claus Michael Ringel
Journal: Trans. Amer. Math. Soc. 274 (1982), 399-443
MSC: Primary 16A46; Secondary 16A64, 18E40
MathSciNet review: 675063
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Abstract: Let $ A$ be a finite dimensional hereditary algebra over a field, with $ n$ simple $ A$-modules. An $ A$-module $ T_A$ with $ n$ pairwise nonisomorphic indecomposable direct summands and satisfying $ {\text{Ex}}{{\text{t}}^1}({T_A},\,{T_A}) = 0$ is called a tilting module, and its endomorphism ring $ B$ is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable $ B$-modules are in one-to-one correspondence to the indecomposable $ A$-modules which are either torsion or torsionfree. One of the main results of the paper asserts that an algebra of finite representation type with an indecomposable sincere representation is a tilted algebra provided its Auslander-Reiten quiver has no oriented cycles. In fact, tilting modules are introduced and studied for any finite dimensional algebra, generalizing recent results of Brenner and Butler.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0675063-2
PII: S 0002-9947(1982)0675063-2
Article copyright: © Copyright 1982 American Mathematical Society