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

Commentationes Mathematicae Universitatis Carolinae (1995)

- Volume: 36, Issue: 3, page 417-421
- ISSN: 0010-2628

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topFernández, 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 -

## References

top- Auslander M., Reiten I., Representation theory of Artin algebras III, Communications in Algebra 3 (1975), 239-294. (1975) Zbl0331.16027MR0379599
- Zimmermann W., Existenz von Auslander-Reiten-Folgen, Archiv der Math. 40 (1983), 40-49. (1983) Zbl0513.16019MR0720892
- Fernández A., Almost split sequences and Morita duality, Bull. des Sciences Math., 2me série, 110 (1986), 425-435. (1986) MR0884217

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