Perfectoid field - Knowlpedia

Definition

A perfectoid field is a complete nonarchimedean field $K$ with a nondiscrete rank-one valuation, residue characteristic $p>0$ , and surjective Frobenius

\varphi:\mathcal O_K/p\longrightarrow\mathcal O_K/p, \qquad x\longmapsto x^p.

In characteristic $p$ , this says that $K$ is perfect in addition to being complete and nondiscretely valued. In mixed characteristic, the condition forces $K$ to contain elements with arbitrarily deep compatible $p$ -power roots in a valuation-theoretic sense.

Examples

The completion of $\mathbb Q_p(p^{1/p^\infty})$ is perfectoid.
The completed algebraic closure $\mathbb C_p$ is perfectoid.
A discretely valued $p$ -adic field is not perfectoid.

Tilt

Every perfectoid field $K$ has a tilt $K^\flat$ of characteristic $p$ . Their absolute Galois groups are canonically isomorphic, and finite étale extensions correspond. The tilting operation is the basic bridge between mixed- and equal-characteristic perfectoid geometry.

References

Peter Scholze, “Perfectoid spaces,” Publications Mathématiques de l'IHÉS 116 (2012), 245–313. arXiv.
Peter Scholze, “Perfectoid spaces: a survey,” in Current Developments in Mathematics 2012, International Press, 2013, 193–227.