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

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 $\Gamma\times\cdots\times\Gamma$ 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 $\Gamma$, thereby generalizing a result of Z\'{a}br\'{a}di (who had treated the case where $\Phi$ is the multiplicative group). We then discuss applications to $(\varphi,\Gamma)$-modules over $k[[t_1,\ldots,t_n]][\prod_it_i^{-1}]$.