论文标题
拓扑光谱和完美的泰特环
Topological spectrum and perfectoid Tate rings
论文作者
论文摘要
我们研究了一个半态环$ r $的拓扑光谱，我们将其定义为Prime Ideals $ \ Mathfrak {p} $的空间，以至于$ \ mathfrak {p} $等于某些有界的功率 - 义务中的eminimorm eminorm的内核。对于任何半态环$ r $，我们表明拓扑光谱是一个准混合清醒的拓扑空间。当$ r $是一个完美的泰特环时，我们在$ r $的拓扑光谱与其倾斜$ r^{\ flat} $的拓扑光谱之间构建了天然同构。作为一个应用程序，我们证明，当且仅当它的倾斜度是一个不可或缺的域时，完美的泰特环$ r $是一个不可或缺的域。
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 $R^{\flat}$. 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.
