Topological spectrum and perfectoid Tate rings

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.