Noetherian properties of Fargues-Fontaine curves

TL;DR

This paper proves that extended Robba rings in p-adic Hodge theory are strongly noetherian Banach rings, enabling the application of Huber's adic space theory and establishing their localizations as principal ideal domains and Dedekind domains.

Contribution

It demonstrates the noetherian properties of extended Robba rings and their localizations, facilitating advanced geometric methods in p-adic Hodge theory.

Findings

Extended Robba rings are strongly noetherian Banach rings.

Rational localizations of these rings are principal ideal domains.

Etale covers of these rings are Dedekind domains.

Abstract

We establish that the extended Robba rings associated to a perfect nonarchimedean field of characteristic p, which arise in p-adic Hodge theory as certain completed localizations of the ring of Witt vectors, are strongly noetherian Banach rings; that is, the completed polynomial ring in any number of variables over such a Banach ring is noetherian. This enables Huber's theory of adic spaces to be applied to such rings. We also establish that rational localizations of these rings are principal ideal domains and that etale covers of these rings (in the sense of Huber) are Dedekind domains.