Tilting and untilting for ideals in perfectoid rings

Abstract

For a 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 in the first author’s previous work. We prove that the two maps

$$\begin{aligned} J\mapsto J^{\flat }~\text {and}~I\mapsto I^{\sharp } \end{aligned}$$

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. Finally, we prove that the maps \((-)^{\flat }\) and \((-)^{\sharp }\) send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of \({{\,\textrm{Spec}\,}}(R)\) consisting of prime ideals \(\mathfrak {p}\) of R such that \(R/\mathfrak {p}\) is perfectoid and the subspace of \({{\,\textrm{Spec}\,}}(R^{\flat })\) 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 work which concerned prime ideals in perfectoid Tate rings.