Submission date: 13 March 2019

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 𝔽 of characteristic p>0, as a valued module over 𝔽[t;x → x^p] is C-minimal. Moreover, any ultraproduct K, of algebraically closed valued fields K_{p^n} of characteristic p>0, endowed each with the morphism x→ x^{p^n}, following a ultrafilter U over {p^n | n in ℕ, et p prime }, equipped with the non-standart Frobenius, i.e., the map σ_U:= lim_U (x → x^{p^n}), is C-minimal as a K[t;σ]-valued module.

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Full text arXiv 1903.05515: pdf, ps.