Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory

Fernández, Ariel

Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 417-421.

Résumé

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.

Détail

Classification : 16D90, 16G70
Keywords: almost split sequence; Morita duality