Timeline for If the tensor product of two $kG$-modules is projeсtive, does either of them have to be projective?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Feb 19, 2011 at 3:11 | vote | accept | Heskie | ||
| Feb 14, 2011 at 4:42 | history | edited | John Palmieri | CC BY-SA 2.5 |
add comment about cyclic sylow p-subgroup
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| Feb 14, 2011 at 0:34 | comment | added | Jim Humphreys |
@John: $SL_2(\mathbb{F}_p)$ is an interesting example to try out in this spirit. Unlike most groups of Lie type, all of its (finitely many) indecomposable modules are known due to the special nature of its Sylow $p$-subgroup, as well as all projective modules. This probably gives the kind of concrete example Heskie is asking for.
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| Feb 13, 2011 at 23:16 | comment | added | John Palmieri | This is a guess, but perhaps if the Sylow $p$-subgroup of $G$ is cyclic of order $p$, or maybe even just cyclic? | |
| Feb 13, 2011 at 23:12 | comment | added | Heskie | Ah, OK. Thank you. I'd already be happy with a (concrete) example of a group $G$ and a field $k$ such that char($k$) divides the order of $G$ and such that $M\otimes_{k}N$ projective, implies $M$ or $N$ projective. | |
| Feb 13, 2011 at 23:06 | history | answered | John Palmieri | CC BY-SA 2.5 |