We study the topological spectrum ${\mathrm{Spec}<!--FUNCTION\; APPLICATION-->}_{\mathrm{Top}<!--FUNCTION\; APPLICATION-->}(R)$ 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 ${\mathrm{Spec}<!--FUNCTION\; APPLICATION-->}_{\mathrm{Top}<!--FUNCTION\; APPLICATION-->}(R)$ is a quasicompact sober topological space. When $R$ is a perfectoid Tate ring we construct a natural homeomorphism

$${\mathrm{Spec}<!--FUNCTION\; APPLICATION-->}_{\mathrm{Top}<!--FUNCTION\; APPLICATION-->}(R)\simeq {\mathrm{Spec}<!--FUNCTION\; APPLICATION-->}_{\mathrm{Top}<!--FUNCTION\; APPLICATION-->}(R^{♭})$$

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：

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拓扑谱和完美型 Tate 环

我们研究拓扑谱${\mathrm{规格}<!--FUNCTION\; APPLICATION-->}_{\mathrm{最佳}<!--FUNCTION\; APPLICATION-->}(R)$半范数环$R$我们将其定义为素理想空间$\mathfrak{𝔭}$这样$\mathfrak{𝔭}$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${\mathrm{规格}<!--FUNCTION\; APPLICATION-->}_{\mathrm{最佳}<!--FUNCTION\; APPLICATION-->}(R)$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

$${\mathrm{规格}<!--FUNCTION\; APPLICATION-->}_{\mathrm{最佳}<!--FUNCTION\; APPLICATION-->}(R)\simeq {\mathrm{规格}<!--FUNCTION\; APPLICATION-->}_{\mathrm{最佳}<!--FUNCTION\; APPLICATION-->}(R^{♭})$$

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。