A note on multivariable $(\phi ,\Gamma )$-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 $k[[t_1,\dots ,t_n]][{\prod }_i{t}_i^{-1}]$ 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 $k[[t_1,\dots ,t_n]][{\prod }_i{t}_i^{-1}]$ admits no non-trivial ideal stable under $\Gamma$, thereby generalizing a result of Zábrádi (who had treated the case where $\Phi$ is the multiplicative group). We then discuss applications to $(\phi ,\Gamma )$-modules over $k[[t_1,\dots ,t_n]][{\prod }_i{t}_i^{-1}]$.

References in corpus (1)

• Multivariable $(\phi ,\Gamma )$-modules and smooth $o$-torsion representations

Cited by in corpus (1)

• Supersingular Hecke modules as Galois representations