Topological spectrum and perfectoid Tate rings

Dine, Dimitri

doi:10.2140/ant.2022.16.1463

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Topological spectrum and perfectoid Tate rings

Dimitri Dine

Vol. 16 (2022), No. 6, 1463–1500

DOI: 10.2140/ant.2022.16.1463

Abstract

We study the topological spectrum ${Spec}_{Top} (R)$ of a seminormed ring $R$ which we define as the space of prime ideals $𝔭$ such that $𝔭$ equals the kernel of some bounded power-multiplicative seminorm. For any seminormed ring $R$ we show that ${Spec}_{Top} (R)$ is a quasicompact sober topological space. When $R$ is a perfectoid Tate ring we construct a natural homeomorphism

{Spec}_{Top} (R) ≃ {Spec}_{Top} (R^{♭})

between the topological spectrum of $R$ and the topological spectrum of its tilt $R^{♭}$ . As an application, we prove that a perfectoid Tate ring $R$ is an integral domain if and only if its tilt is an integral domain.

Keywords

perfectoid rings, tilting equivalence, Tate rings, Banach rings

Mathematical Subject Classification

Primary: 13J99, 14G22, 14G45

Milestones

Received: 14 November 2020

Revised: 5 July 2021

Accepted: 5 August 2021

Published: 27 September 2022

Authors

Dimitri Dine
Fakultät für Mathematik Technische Universität München München Germany

Vol. 16, No. 6, 2022