Timeline for $\mathrm{Ext}$ group in representation theory
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 28, 2017 at 9:19 | answer | added | Matthew Pressland | timeline score: 1 | |
| Jul 28, 2017 at 9:14 | comment | added | Matthew Pressland | Sure: in that case you are essentially working with a bunch of copies of $k\mathcal{Q}$ (although possibly for different fields $k$). (I don't actually have a counterexample to the general statement, but if it is true it has to be for a quite different reason, since it is certainly possible for the global dimension of $R\mathcal{Q}$ to be greater than $1$ in general.) | |
| Jul 28, 2017 at 7:55 | comment | added | Homa81 | Thank you so much. So if we add the condition that $R$ is a semisimple ring we can prove the claim. Is it true? | |
| Jul 27, 2017 at 15:49 | history | edited | Homa81 | CC BY-SA 3.0 |
edited body
|
| Jul 27, 2017 at 14:56 | comment | added | Matthew Pressland | This is true when $R$ is a field, by applying $\operatorname{Hom}_{\text{rep}}(-,R\mathcal{Q})$ to the short exact sequence $0\to\mathcal{X}'\to\mathcal{X}\to C\to0$, which exists since $v_1$ is a source, and using that $RQ$ has global dimension $1$ in this case. In the general case, it is true (by the same argument) if and only if $\operatorname{Ext}^i_{\text{rep}}(C,R\mathcal{Q})=0$ for all $i\geq2$. I suspect this is false, but I don't have a counterexample immediately. | |
| Jul 27, 2017 at 12:42 | history | asked | Homa81 | CC BY-SA 3.0 |