Topological spectrum and perfectoid Tate rings (Q2676962)

Let \(R\) be a seminormed ring and let \(J \subsetneq A\) be an ideal. Then \(J\) is called \textit{spectrally reduced} if there exists a power-multiplicative bounded seminorm \(\phi\) on \(A\) such that \(\text{ker }\phi = J\). Moreover, the \textit{topological spectrum} of \(A\) is the subspace \[ \mathrm{Spec}_{\mathrm{Top}}(A) = \{ \mathfrak{p} \in \mathrm{Spec}(A)~|~\mathfrak{p}\text{ is spectrally reduced}\}. \] of Spec(\(A\)) endowed with the subspace topology induced by the Zariski topology on Spec(\(A\)). The main result in this paper is the existence of the the homeomorphism between the topological spectrum of a perfectoid Tate ring \(A\) and that of its tilt \(A^\flat\). More precisely, he proved that there is a following homeomorphism for Zariski topology on \(\mathrm{Spec}_{\mathrm{Top}}(A)\) and \(\mathrm{Spec}_{\mathrm{Top}}(A^\flat)\) \[ \mathrm{Spec}_{\mathrm{Top}}(A) \to\mathrm{Spec}_{\mathrm{Top}}(A^\flat) ~;~ \mathfrak{p} \mapsto \mathfrak{p}^\flat = \{ (f^{(n)})_{n\geq 0}\in R^\flat~|~f^{(0)} \in \mathfrak{p} \}, \] and its inverse \(\mathfrak{q} \mapsto \mathfrak{q}^{\sharp}\) is formulated using the homeomorphism of the Berkovich spectrum of \(A\) and that of its tilt \(A^\flat\). As a corollary, the author showed a perfectoid Tate ring \(A\) is an integral domain if and only if its tilt \(A^\flat\) is also an integral domain.