A note on multivariable \((\varphi ,\Gamma )\)-modules

  • Elmar Grosse-KlönneEmail author


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 \(k[[t_1,\ldots ,t_n]][\prod _it_i^{-1}]\).



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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© SpringerNature 2018

Authors and Affiliations

  1. 1.Humboldt-Universität zu Berlin, Institut für MathematikBerlinGermany

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