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Title: Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory (English)
Author: Fernández, Ariel
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 36
Issue: 3
Year: 1995
Pages: 417-421
Category: math
Summary: We take a complementary view to the Auslander-Reiten trend of thought: Instead of specializing a module category to the level where the existence of an almost split sequence is inferred, we explore the structural consequences that result if we assume the existence of a single almost split sequence under the most general conditions. We characterize the structure of the bimodule ${{}_{\Delta }\!}\operatorname Ext {}_{R}(C,A)_{\Gamma }$ with an underlying ring $R$ solely assuming that there exists an almost split sequence of left $R$-modules $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. $\Delta $ and $\Gamma $ are quotient rings of $\operatorname End({}_{R} C)$ and $\operatorname End({}_{R} A)$ respectively. The results are dualized under mild assumptions warranting that ${{}_{\Delta }\!}\operatorname Ext {}_{R}(C,A)_{\Gamma }$ represent a Morita duality. To conclude, a reciprocal result is obtained: Conditions are imposed on ${{}_{\Delta }\!}\operatorname Ext {}_{R}(C,A)_{\Gamma }$ that warrant the existence of an almost split sequence. (English)
Keyword: almost split sequence
Keyword: Morita duality
MSC: 16D90
MSC: 16G70
idZBL: Zbl 0839.16013
idMR: MR1364480
Date available: 2009-01-08T18:18:53Z
Last updated: 2012-04-30
Stable URL:
Reference: [1] Auslander M., Reiten I.: Representation theory of Artin algebras III.Communications in Algebra 3 (1975), 239-294. Zbl 0331.16027, MR 0379599
Reference: [2] Zimmermann W.: Existenz von Auslander-Reiten-Folgen.Archiv der Math. 40 (1983), 40-49. Zbl 0513.16019, MR 0720892
Reference: [3] Fernández A.: Almost split sequences and Morita duality.Bull. des Sciences Math., 2me série, 110 (1986), 425-435. MR 0884217


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