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# Article

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Keywords:
almost split sequence; Morita duality
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.
References:
[1] Auslander M., Reiten I.: Representation theory of Artin algebras III. Communications in Algebra 3 (1975), 239-294. MR 0379599 | Zbl 0331.16027
[2] Zimmermann W.: Existenz von Auslander-Reiten-Folgen. Archiv der Math. 40 (1983), 40-49. MR 0720892 | Zbl 0513.16019
[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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