Tilting and untilting for ideals in perfectoid rings
Dimitri Dine, Ryo Ishizuka
TL;DR
The paper develops a purely algebraic tilting/untilting framework for perfectoid rings of characteristic $0$, establishing a bijection between perfectoid ideals $J$ of $R$ and $p^{\flat}$-adically closed radical ideals of $R^{\flat}$ via $(-)^{\flat}$ and $(-)^{\sharp}$. It proves that tilting sends $p$-adically closed prime ideals to primes in $R^{\flat}$ and untilting preserves primeness, yielding a homeomorphism between the perfectoid spectra $\mathop{Spec}_{\mathrm{perfd}}(R)$ and $\mathop{Spec}_{\mathrm{perfd}}(R^{\flat})$. The results generalize earlier work on perfectoid Tate rings and provide a purely algebraic proof of the prime-ideals correspondence without relying on Berkovich or adic spectrum homeomorphisms. The framework clarifies how perfectoid quotients behave under tilting and extends to general perfectoid rings, offering tools for $p$-adic geometry and commutative algebra. Overall, the paper unifies the ideal-theoretic and spectral aspects of tilting in the perfectoid setting and broadens the scope beyond 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.