Tilting and untilting for ideals in perfectoid rings

TL;DR

This paper introduces algebraic tilting and untilting maps between ideals in perfectoid rings and their tilts, establishing a bijection and homeomorphism that generalize previous results and deepen understanding of ideal structures in perfectoid theory.

Contribution

It defines algebraic tilting and untilting maps for ideals in perfectoid rings, extending previous analytic concepts and establishing a bijective correspondence and homeomorphism between certain ideal spectra.

Findings

Establishes a bijection between ideals with perfectoid quotients and radical ideals in the tilt.

Shows the maps send prime ideals to prime ideals, inducing a homeomorphism.

Generalizes and provides a new proof for the main result on prime ideals in perfectoid Tate rings.

Abstract

For an (integral) perfectoid ring $R$ of characteristic $0$ with tilt $R^{\flat}$, we introduce and study a tilting map $(-)^{\flat}$ from the set of $p$-adically closed ideals of $R$ to the set of ideals of $R^{\flat}$ and an untilting map $(-)^{\sharp}$ from the set of radical ideals of $R^{\flat}$ to the set of ideals of $R$. The untilting map $(-)^{\sharp}$ is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic $p$ introduced by the first author. We prove that these two maps, $(-)^{\flat}$ and $(-)^{\sharp}$, define an inclusion-preserving bijection between the set of ideals $J$ of $R$ such that the quotient $R/J$ is perfectoid and the set of $p^{\flat}$-adically closed radical ideals of $R^{\flat}$, where $p^{\flat}\in R^{\flat}$ corresponds to a compatible system of $p$-power roots of a unit multiple of $p$ in $R$. Furthermore, we prove that the maps send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of the spectrum of $R$ consisting of prime ideals $\mathfrak{p}$ of $R$ such that $R/\mathfrak{p}$ is perfectoid and the subspace of the spectrum of $R^{\flat}$ consisting of $p^{\flat}$-adically closed prime ideals of $R^{\flat}$. In particular, we obtain a generalization and a new proof of the main result of the first author's previous research which concerned prime ideals in perfectoid Tate rings.