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2022 Topological spectrum and perfectoid Tate rings
Dimitri Dine
Algebra Number Theory 16(6): 1463-1500 (2022). DOI: 10.2140/ant.2022.16.1463

Abstract

We study the topological spectrum SpecTop(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 SpecTop(R) is a quasicompact sober topological space. When R is a perfectoid Tate ring we construct a natural homeomorphism

SpecTop(R)SpecTop(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.

Citation

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Dimitri Dine. "Topological spectrum and perfectoid Tate rings." Algebra Number Theory 16 (6) 1463 - 1500, 2022. https://doi.org/10.2140/ant.2022.16.1463

Information

Received: 14 November 2020; Revised: 5 July 2021; Accepted: 5 August 2021; Published: 2022
First available in Project Euclid: 2 October 2022

Digital Object Identifier: 10.2140/ant.2022.16.1463

Subjects:
Primary: 13J99 , 14G22 , 14G45

Keywords: Banach rings , perfectoid rings , Tate rings , tilting equivalence

Rights: Copyright © 2022 Mathematical Sciences Publishers

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Vol.16 • No. 6 • 2022
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