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My question arises from the previous question Let $M$ be a finitely generated $\mathbb{Q}[\mathbb{Z}/p^l]$-module, where $p$ is a prime number. Is it true that \begin{equation}\dim_\mathbb{Q} M\geq \frac{p}{p-1}\dim_\mathbb{Q} M \mathbin{\mathop{\otimes}_{\mathbb{Q}[\mathbb{Z}/p^l]}}\mathbb{Q}(\zeta_{p^l})?\end{equation} Here, $\zeta_{p^l}$ is the $p^l$th root of the unity. |
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Your ring is the direct sum of fields $$ \frac{\mathbb{Q}[x]}{x^{p^{\ell}} - 1} = \bigoplus_{i=0}^{\ell} \mathbb{Q}(\zeta_{\ell^i}), $$ and a module over your ring is just a direct sum of vector spaces over these fields; the tensor product functor you wrote just projects such a module to the $\mathbb{Q}(\zeta_{p^\ell})$-vector-space-component. The inequality is false, because you can just take $M = \mathbb{Q}(\zeta_{p^\ell})$ itself. Maybe you don't want $\mathbb{Q}$-dimension on both sides? |
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