Toute forme modérément ramifiée d’un polydisque ouvert est triviale

Abstract

Let k be a complete, non-Archimedean field and let X be a k-analytic space. Assume that there exists a finite, tamely ramified extension L of 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 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 étale algebras, etc.