Timeline for Existence of a certain direct summand
Current License: CC BY-SA 4.0
5 events
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| Jul 24, 2018 at 11:48 | comment | added | debanjana | Thanks. These are both good starting points! I am/ was hoping to find (weak) conditions on say the $\dim_{\mathbb{Q}_p}M_p$ to say something about $m_d$? I wasn't sure if it is some standard result that I am unaware of. | |
| Jul 24, 2018 at 10:38 | comment | added | Frieder Ladisch | One (necessary and sufficient) condition is that the character of $M_p$ contains every linear character of $P$ with multiplicity $>0$. In your notation, if $m_d>0$ for all $d \mid p^n$. This is so since $\mathbb{Q}_p[P]$ is semisimple, and in fact $\mathbb{Q}_p[P]\cong \bigoplus_{d\mid p^n} \mathbb{Q}_p(\zeta_d)$. Is this the kind of condition you're looking for? | |
| Jul 24, 2018 at 9:41 | comment | added | Steffen Kionke | Do you want a condition on $M$ or on $M_p$? I think, if $M$ is projective as $\mathbb{Z}_p[P]$ modules, then $M_p$ is free. On the other hand, if you only care about $M_p$, then the existence of $M$ does not help you. Any finite dimensional $\mathbb{Q}_p$ representation of $P$ contains $P$-stable $\mathbb{Z}_p$ lattice. | |
| Jul 24, 2018 at 5:25 | history | edited | Martin Sleziak |
Removed the deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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| Jul 23, 2018 at 22:22 | history | asked | debanjana | CC BY-SA 4.0 |