Toute forme mod\'er\'ement ramifi\'ee d'un polydisque ouvert est triviale

TL;DR

The paper proves that a tamely ramified form of an open polydisc over a non-Archimedean field is trivial, showing it is isomorphic to an open polydisc over the base field using graded reduction techniques.

Contribution

It introduces a method using graded reduction and classical algebra results to establish the triviality of tamely ramified forms of open polydiscs over non-Archimedean fields.

Findings

Tamely ramified forms of open polydiscs are trivial over the base field.

Uses graded reduction to analyze algebra of functions on non-Archimedean spaces.

Provides a new approach to understanding forms of analytic spaces in non-Archimedean geometry.

Abstract

Let k be a complete, non-Archimedean field and let X be a k-analytic space ; assume that there exists a tamely ramified finite extension L/k such that X_L is isomorphic to an open polydisc over L ; we prove that X is itself isomorphic to an open polydisc over k. The proof consists in using the {\em graded} reduction (a notion which is due to Temkin) of the algebra of functions on $X$, together with some graded counterparts of classical commutative algebra results: Nakayama's lemma, going-up theorem, basic notions about \'etale algebras, etc.