Lemma 59.66.3. Let $\Lambda $ be a Noetherian ring. Let $\ell $ be a prime number and $n \geq 1$. Let $H$ be a finite $\ell $-group. Let $M$ be a finite $\Lambda [H]$-module annihilated by $\ell ^ n$. Then there is a finite filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_ t = M$ by $\Lambda [H]$-submodules such that $H$ acts trivially on $M_{i + 1}/M_ i$ for all $i = 0, \ldots , t - 1$.
Proof. Omitted. Hint: Show that the augmentation ideal $\mathfrak m$ of the noncommutative ring $\mathbf{Z}/\ell ^ n\mathbf{Z}[H]$ is nilpotent. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: