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

Fernández, Ariel. "Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 417-421. <http://eudml.org/doc/247760>.

@article{Fernández1995,
abstract = {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 $\{\{\}_\{\Delta \}\!\}\operatorname\{E\}xt \{\}_\{R\}(C,A)_\{\Gamma \}$ 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 $\operatorname\{E\}nd(\{\}_\{R\} C)$ and $\operatorname\{E\}nd(\{\}_\{R\} A)$ respectively. The results are dualized under mild assumptions warranting that $\{\{\}_\{\Delta \}\!\}\operatorname\{E\}xt \{\}_\{R\}(C,A)_\{\Gamma \}$ represent a Morita duality. To conclude, a reciprocal result is obtained: Conditions are imposed on $\{\{\}_\{\Delta \}\!\}\operatorname\{E\}xt \{\}_\{R\}(C,A)_\{\Gamma \}$ that warrant the existence of an almost split sequence.},
author = {Fernández, Ariel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {almost split sequence; Morita duality; almost split sequences; Auslander-Reiten sequences; module category; existence; Morita duality},
language = {eng},
number = {3},
pages = {417-421},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory},
url = {http://eudml.org/doc/247760},
volume = {36},
year = {1995},
}

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
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 3
SP - 417
EP - 421
AB - 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 ${{}_{\Delta }\!}\operatorname{E}xt {}_{R}(C,A)_{\Gamma }$ 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 $\operatorname{E}nd({}_{R} C)$ and $\operatorname{E}nd({}_{R} A)$ respectively. The results are dualized under mild assumptions warranting that ${{}_{\Delta }\!}\operatorname{E}xt {}_{R}(C,A)_{\Gamma }$ represent a Morita duality. To conclude, a reciprocal result is obtained: Conditions are imposed on ${{}_{\Delta }\!}\operatorname{E}xt {}_{R}(C,A)_{\Gamma }$ that warrant the existence of an almost split sequence.
LA - eng
KW - almost split sequence; Morita duality; almost split sequences; Auslander-Reiten sequences; module category; existence; Morita duality
UR - http://eudml.org/doc/247760
ER -