| Title:
|
Almost split sequences and module categories: A complementary view to Auslander-Reiten Theory (English) |
| Author:
|
Fernández, Ariel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
36 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
417-421 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
almost split sequence |
| Keyword:
|
Morita duality |
| MSC:
|
16D90 |
| MSC:
|
16G70 |
| idZBL:
|
Zbl 0839.16013 |
| idMR:
|
MR1364480 |
| . |
| Date available:
|
2009-01-08T18:18:53Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118768 |
| . |
| Reference:
|
[1] Auslander M., Reiten I.: Representation theory of Artin algebras III.Communications in Algebra 3 (1975), 239-294. Zbl 0331.16027, MR 0379599 |
| Reference:
|
[2] Zimmermann W.: Existenz von Auslander-Reiten-Folgen.Archiv der Math. 40 (1983), 40-49. Zbl 0513.16019, MR 0720892 |
| Reference:
|
[3] Fernández A.: Almost split sequences and Morita duality.Bull. des Sciences Math., 2me série, 110 (1986), 425-435. MR 0884217 |
| . |
