Tilting and untilting for ideals in perfectoid rings

Abstract

AbstractFor a perfectoid ring R of characteristic 0 with tilt $$R^{\flat }$$ R ♭ , we introduce and study a tilting map $$(-)^{\flat }$$ ( - ) ♭ from the set of p-adically closed ideals of R to the set of ideals of $$R^{\flat }$$ R ♭ and an untilting map $$(-)^{\sharp }$$ ( - ) ♯ from the set of radical ideals of $$R^{\flat }$$ R ♭ 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 $$\begin{aligned} J\mapsto J^{\flat }~\text {and}~I\mapsto I^{\sharp } \end{aligned}$$ J ↦ J ♭ and I ↦ I ♯ 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 $$p^{\flat }$$ p ♭ -adically closed radical ideals of $$R^{\flat }$$ R ♭ , where $$p^{\flat }\in R^{\flat }$$ p ♭ ∈ R ♭ 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 $${{\,\textrm{Spec}\,}}(R)$$ Spec ( R ) consisting of prime ideals $$\mathfrak {p}$$ p of R such that $$R/\mathfrak {p}$$ R / p is perfectoid and the subspace of $${{\,\textrm{Spec}\,}}(R^{\flat })$$ Spec ( R ♭ ) consisting of $$p^{\flat }$$ p ♭ -adically closed prime ideals of $$R^{\flat }$$ R ♭ . 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.