#### Vol. 16, No. 6, 2022

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Topological spectrum and perfectoid Tate rings

### Dimitri Dine

Vol. 16 (2022), No. 6, 1463–1500
##### Abstract

We study the topological spectrum ${\mathrm{Spec}}_{\mathrm{Top}}\left(R\right)$ 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 ${\mathrm{Spec}}_{\mathrm{Top}}\left(R\right)$ is a quasicompact sober topological space. When $R$ is a perfectoid Tate ring we construct a natural homeomorphism

 ${\mathrm{Spec}}_{\mathrm{Top}}\left(R\right)\simeq {\mathrm{Spec}}_{\mathrm{Top}}\left({R}^{♭}\right)$

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.

##### Keywords
perfectoid rings, tilting equivalence, Tate rings, Banach rings
##### Mathematical Subject Classification
Primary: 13J99, 14G22, 14G45