Modules C-minimaux sur des anneaux de polynômes tordus

Abstract

In this article we study modules endowed with a ultrametric, from the point of view of the geometric notion $C$-minimality. We give a complete characterization of $C$-minimal valued modules over non-commutative rings of skew polynomials of the form $R:=K[t;φ]$, where $K$ is a field, $φ$ an endomorphism of $K$ and $R$ is the $K$-algebra generated by $t$, such that $at=ta^φ$ for $a\in K$. We deduce for instance that the ring of Puiseux series over a finite field $\mathbb{F}$ of characteristic $p>0$, as a valued module over $\mathbb{F}[t;x\mapsto x^p]$ is $C$-minimal. Moreover, any ultraproduct $\mathcal{K}$, of algebraically closed valued fields $\mathcal{K}_{p^n}$ of characteristic $p>0$, endowed each with the morphism $x\mapsto x^{p^n}$, following a ultrafilter $U$ over $\{p^n\ |\ n\in \mathbb{N}, \, \text{et} \; p \; \text{prime}\}$, equipped with the {\it non-standart Frobenius}, i.e., the map $σ_{U}:=\lim_{U} x \mapsto x^{p^n}$, is $C$-minimal as a $\mathcal{K}[t;σ]$-valued module.