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Topological spectrum and perfectoid Tate rings
Algebra & Number Theory ( IF 0.9 ) Pub Date : 2022-09-27 , DOI: 10.2140/ant.2022.16.1463
Dimitri Dine

We study the topological spectrum Spec Top (R) of a seminormed ring R which we define as the space of prime ideals 𝔭 such that 𝔭 equals the kernel of some bounded power-multiplicative seminorm. For any seminormed ring R we show that Spec Top (R) is a quasicompact sober topological space. When R is a perfectoid Tate ring we construct a natural homeomorphism

Spec Top (R) Spec Top (R)

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.



中文翻译:

拓扑谱和完美型 Tate 环

我们研究拓扑谱规格 最佳 (R)半范数环R我们将其定义为素理想空间𝔭这样𝔭等于某个有界幂乘半范数的核。对于任何半规范环R我们证明了规格 最佳 (R)是一个拟紧致清醒拓扑空间。什么时候R是一个完美的 Tate 环 我们构造一个自然同胚

规格 最佳 (R) 规格 最佳 (R)

的拓扑谱之间R及其倾斜的拓扑谱R. 作为一个应用,我们证明了一个完美的 Tate 环R是一个积分域当且仅当它的倾斜是一个积分域。

更新日期:2022-09-28
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