The Stacks project

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. $\square$

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