Abstract
Let \(F/{\mathbb Q}_p\) be a finite field extension, let k be a field of characteristic p. Fix a Lubin Tate group \(\Phi \) for F and let \(\Gamma _{\bullet }=\Gamma \times \cdots \times \Gamma \) with \(\Gamma ={\mathcal O}_F^{\times }\) act on \(k[[t_1,\ldots ,t_n]][\prod _it_i^{-1}]\) by letting \(\gamma _i\) (in the i-th factor \(\Gamma \)) act on \(t_i\) by insertion of \(t_i\) into the power series attached to \(\gamma _i\) by \(\Phi \). We show that \(k[[t_1,\ldots ,t_n]][\prod _it_i^{-1}]\) admits no non-trivial ideal stable under \(\Gamma _{\bullet }\), thereby generalizing a result of Zábrádi (who had treated the case where \(\Phi \) is the multiplicative group). We then discuss applications to \((\varphi ,\Gamma )\)-modules over \(k[[t_1,\ldots ,t_n]][\prod _it_i^{-1}]\).
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Notes
Notice that we do not require \(\psi _{d}(1)=1\).
For a \(k[[t_{\bullet }]]\)-module X, resp. a \(k((t_{\bullet }))\)-module X, we call \(\mathrm{dim}_{\mathrm{Frac}(k[[t_{\bullet }]])}X\otimes _{k[[t_{\bullet }]]} \mathrm{Frac}(k[[t_{\bullet }]])\), resp. \(\mathrm{dim}_{\mathrm{Frac}(k[[t_{\bullet }]])}X\otimes _{k((t_{\bullet }))} \mathrm{Frac}(k[[t_{\bullet }]])\), the generic rank of X, where \(\mathrm{Frac}(k[[t_{\bullet }]])\) denotes the fraction field of \(k[[t_{\bullet }]]\).
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Acknowlegements
I thank Gergely Zábrádi for his careful reading of an earlier draft of the proof of Theorem 1 and for further discussions on the topic. I thank the referees for their valuable comments.
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Grosse-Klönne, E. A note on multivariable \((\varphi ,\Gamma )\)-modules. Res. number theory 5, 6 (2019). https://doi.org/10.1007/s40993-018-0144-8
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DOI: https://doi.org/10.1007/s40993-018-0144-8