Tilting and untilting for ideals in perfectoid rings

For a perfectoid ring R of characteristic 0 with tilt R ♭ , we introduce and study a tilting map ( - ) ♭ from the set of p -adically closed ideals of R to the set of ideals of R ♭ and an untilting map ( - ) ♯ from the set of radical ideals of R ♭ to the set of ideals of R . The untilting map ( - ) ♯ 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 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 ♭ -adically closed radical ideals of R ♭ , where 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 ( - ) ♭ and ( - ) ♯ send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of Spec ( R ) consisting of prime ideals p of R such that R / p is perfectoid and the subspace of Spec ( R ♭ ) consisting of p ♭ -adically closed prime ideals of 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.