Topological spectrum and perfectoid Tate rings

TL;DR

This paper investigates the topological spectrum of seminormed rings, especially perfectoid Tate rings, establishing a homeomorphism with their tilt's spectrum and characterizing when such rings are integral domains.

Contribution

It introduces the topological spectrum for seminormed rings and demonstrates a natural homeomorphism between spectra of perfectoid Tate rings and their tilts, with applications to integral domain characterization.

Findings

The topological spectrum is a quasi-compact sober space for any seminormed ring.

A homeomorphism exists between the spectra of perfectoid Tate rings and their tilts.

A perfectoid Tate ring is an integral domain if and only if its tilt is an integral domain.

Abstract

We study the topological spectrum of a seminormed ring $R$ which we define as the space of prime ideals $\mathfrak{p}$ such that $\mathfrak{p}$ equals the kernel of some bounded power-multiplicative seminorm. For any seminormed ring $R$ we show that the topological spectrum is a quasi-compact sober topological space. When $R$ is a perfectoid Tate ring we construct a natural homeomorphism between the topological spectrum of $R$ and the topological spectrum of its tilt $R^{\flat}$. 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.