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.
@article{118768,
author = {Ariel Fern\'andez},
title = {Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {36},
year = {1995},
pages = {417-421},
zbl = {0839.16013},
mrnumber = {1364480},
language = {en},
url = {http://dml.mathdoc.fr/item/118768}
}
Fernández, Ariel. Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) pp. 417-421. http://gdmltest.u-ga.fr/item/118768/
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