Timeline for Auslander-Reiten sequence and projective covers
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 25, 2020 at 18:29 | comment | added | Jianrong Li | @JeremyRickard, thank you very much for your answer. | |
| Oct 25, 2020 at 17:56 | comment | added | Jeremy Rickard | I don't think the same argument works, as I think the only AR-sequences there are for the category of Cohen-Macaulay modules, which doesn't include the simple modules. | |
| Oct 25, 2020 at 15:52 | comment | added | Jianrong Li | @JeremyRickard, thank you very much for your help. I have another question. I think that this property is true for the Auslander-Reiten sequence of the algebra $B_{k,n}$ in the post. Does your proof also work for the algebra $B_{k,n}$? | |
| Oct 25, 2020 at 8:32 | comment | added | Jianrong Li | @JeremyRickard, thank you very much for your proof. | |
| Oct 24, 2020 at 21:00 | comment | added | Jeremy Rickard | The question is equivalent to asking whether the head of $B$ is isomorphic to the direct sum of the heads of $A$ and $C$, which can be detected by applying $\operatorname{Hom}_R(-.S)$ to the sequence for each simple module $S$, and by the definition of an Auslander-Reiten sequence, that gives a short exact sequence if and only if $A\not\cong S$. | |
| Oct 24, 2020 at 19:16 | comment | added | Jianrong Li | @JeremyRickard, thank you very much. Are there some references about this fact? I need to cite this fact in a paper. | |
| Oct 24, 2020 at 17:26 | comment | added | Jeremy Rickard | Did you look at any examples? It's true if and only if $A$ is not simple, so you would have found a counterexample by looking at literally any Auslander-Reiten quiver (of a non-semisimple algebra). | |
| Oct 24, 2020 at 16:05 | history | edited | Jianrong Li | CC BY-SA 4.0 |
added 331 characters in body
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| Oct 24, 2020 at 15:12 | answer | added | Mare | timeline score: 3 | |
| Oct 24, 2020 at 14:59 | history | asked | Jianrong Li | CC BY-SA 4.0 |