Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory
Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 417-421
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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.
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{CMUC_1995_36_3_a0,
author = {Fern\'andez, Ariel},
title = {Almost split sequences and module categories: {A} complementary view to {Auslander-Reiten} {Theory}},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {417--421},
year = {1995},
volume = {36},
number = {3},
mrnumber = {1364480},
zbl = {0839.16013},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_1995_36_3_a0/}
}
TY - JOUR AU - Fernández, Ariel TI - Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory JO - Commentationes Mathematicae Universitatis Carolinae PY - 1995 SP - 417 EP - 421 VL - 36 IS - 3 UR - http://geodesic.mathdoc.fr/item/CMUC_1995_36_3_a0/ LA - en ID - CMUC_1995_36_3_a0 ER -
%0 Journal Article %A Fernández, Ariel %T Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory %J Commentationes Mathematicae Universitatis Carolinae %D 1995 %P 417-421 %V 36 %N 3 %U http://geodesic.mathdoc.fr/item/CMUC_1995_36_3_a0/ %G en %F CMUC_1995_36_3_a0
Fernández, Ariel. Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 417-421. http://geodesic.mathdoc.fr/item/CMUC_1995_36_3_a0/