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 $R^{♭}$. 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.

38 pages; changes in the proof of Theorem 2.23, corrected typos, updated acknowledgements and references

References in corpus (1)

• An Open Mapping Theorem for rings which have a zero sequence of units

Cited by in corpus (2)

• A calculation of the perfectoidization of semiperfectoid rings
• Tilting and untilting for ideals in perfectoid rings