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

Ariel Fernández

Commentationes Mathematicae Universitatis Carolinae (1995)

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

Abstract

top
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 Δ E x t R ( C , A ) Γ with an underlying ring R solely assuming that there exists an almost split sequence of left R -modules 0 A B C 0 . Δ and Γ are quotient rings of E n d ( R C ) and E n d ( R A ) respectively. The results are dualized under mild assumptions warranting that Δ E x t R ( C , A ) Γ represent a Morita duality. To conclude, a reciprocal result is obtained: Conditions are imposed on Δ E x t R ( C , A ) Γ that warrant the existence of an almost split sequence.

How to cite

top

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 -

References

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.