Vol. 16, No. 6, 2022

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Topological spectrum and perfectoid Tate rings

Dimitri Dine

Vol. 16 (2022), No. 6, 1463–1500

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.

perfectoid rings, tilting equivalence, Tate rings, Banach rings
Mathematical Subject Classification
Primary: 13J99, 14G22, 14G45
Received: 14 November 2020
Revised: 5 July 2021
Accepted: 5 August 2021
Published: 27 September 2022
Dimitri Dine
Fakultät für Mathematik
Technische Universität München