Formes mod\'er\'ement ramifi\'ees de polydisques ferm\'es et de dentelles

TL;DR

This paper characterizes certain non-Archimedean analytic spaces as polydiscs or laces based on their behavior over tamely ramified Galois extensions, linking their structure to Galois actions.

Contribution

It provides a criterion for identifying moderate ramified forms of polydiscs and laces via Galois descent and the notion of reasonable Galois action.

Findings

Spaces are isomorphic to polydiscs or laces if their base change to an extension has a reasonable Galois action.

The work establishes a correspondence between the structure of $k$-analytic spaces and their extensions.

Galois descent techniques are used to classify these non-Archimedean spaces.

Abstract

Let $k$ be a complete non-Archimedean field, $L$ a finite tamely ramified galoisian extension of $k$ and $X$ a $k$-analytic space. We show that $X$ is isomorphic to a closed $k$-polydisc (resp. a $k$-lace) if and only if $X_L$ is isomorphic to a closed $L$-polydisc (resp. a $L$-lace) on which the action of $\mathrm{Gal}(L/k)$ is reasonable.