Tilting and untilting for ideals in perfectoid rings

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 idea...

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Veröffentlicht in:arXiv.org 2023-08
Hauptverfasser: Dine, Dimitri, Ishizuka, Ryo
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:2331-8422