Topological spectrum and perfectoid Tate rings

Abstract

We study the topological spectrum of a semi-normed 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 semi-norm. For any semi-normed ring \(R\) we show that the topological spectrum is a spectral space in the sense of Hochster. 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.