Timeline for Indecomposable representations for group ring $RG$ over commutative ring $R$ with characteristic $p$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 3, 2021 at 9:54 | vote | accept | Master Gang | ||
| Aug 26, 2021 at 0:29 | comment | added | Will Sawin | @MasterGang Maybe a PID or a Dedekind domain would be a better start. I think Mare's example shows $R= K[x,y]$ is impossible, so UFDs of dimension at least two are hopeless. | |
| Aug 26, 2021 at 0:19 | comment | added | Master Gang | @Will Sawin Maybe I should first think the case that $R$ is UFD with characteristic $p$. | |
| Aug 25, 2021 at 16:03 | answer | added | Mare | timeline score: 2 | |
| Aug 25, 2021 at 14:59 | comment | added | Will Sawin | Your question is at least as hard as classifying the indecomposable modules of an arbitrary commutative ring $R$ but also, since $R[G]$ is a commutative ring, is a special case of classifying the indecomposable modules of an arbitrary ring $R$. So this is more a question about rings than groups. Even if you put conditions on $R$, as Johannes suggests, note that $R[G]=R[u]/(u^{p^n}-1)= R[x]/ ( (1+x)^{p^n}-1 ) = R[x]/ (x^{p^n})$ so $R[G]$ is fairly similar to $R$, so you will need the conditions to be very stringent for this to be more of a question about $G$ than a question about $R$. | |
| Aug 25, 2021 at 14:53 | comment | added | Johannes Hahn | You may want to add some conditions on $R$. Otherwise all the weird complications $R$-modules can have, will potentially be present in $RG$-modules as well. For example: If you have non-trivial idempotents in $R$, you have "unexpected" central idempotents ins $RG$ that further decompose every module that you may have expected to be indecomposable. | |
| Aug 25, 2021 at 14:22 | history | asked | Master Gang | CC BY-SA 4.0 |