Tilt and untilt of a perfectoid field

Definition

Let $K$ be a perfectoid field of residue characteristic $p$ . Its tilt is the characteristic- $p$ field whose ring of integers has multiplicative presentation

\mathcal O_{K^\flat} =\varprojlim_{x\mapsto x^p}\mathcal O_K/p.

Addition is reconstructed from the inverse-limit multiplication. The map $x\mapsto x^\sharp$ sends a compatible sequence to its multiplicative limit in $K$ ; it is multiplicative but generally not additive.

An untilt of a perfectoid field $L$ of characteristic $p$ is a perfectoid field $K$ , often of characteristic $0$ , equipped with an identification $K^\flat\simeq L$ . A field can have many nonisomorphic untilts.

Tilting equivalence

Tilting induces equivalences between finite étale extensions of $K$ and $K^\flat$ , hence an isomorphism of their absolute Galois groups. It also induces an equivalence between perfectoid spaces over the two fields and identifies their underlying topological spaces and étale sites.

Untilts and the Fargues–Fontaine curve

Primitive ideals in period rings, or equivalently suitable points of the Fargues–Fontaine curve, encode untilts of a fixed characteristic- $p$ perfectoid field. This turns the curve into a parameter space for characteristic-zero realizations.

References

Peter Scholze, “Perfectoid spaces,” Publications Mathématiques de l'IHÉS 116 (2012), 245–313. arXiv.
Laurent Fargues and Jean-Marc Fontaine, Courbes et fibrés vectoriels en théorie de Hodge $p$ -adique, Astérisque 406, 2018.