Perfectoid space - Knowlpedia

Definition

Let $K$ be a perfectoid field. A Banach $K$ -algebra $R$ is perfectoid if it is uniform and Frobenius is surjective on $R^\circ/p$ . An affinoid perfectoid space is an adic space

\operatorname{Spa}(R,R^+)

for a perfectoid $K$ -algebra $R$ and an open integrally closed $R^+\subseteq R^\circ$ . A perfectoid space is an adic space locally of this form.

Equivalent definitions replace $p$ by a pseudo-uniformizer whose $p$ th power divides $p$ ; this is useful over general perfectoid bases.

Tilting equivalence

Tilting sends a perfectoid space $X/K$ to a perfectoid space $X^\flat/K^\flat$ with the same underlying topological space. It identifies étale sites and finite étale covers, although it does not identify the structure sheaves as ordinary rings.

Role in local Langlands geometry

Perfectoid spaces underlie diamonds, the v-topology, the Fargues–Fontaine curve, and infinite-level local Shimura varieties. They allow towers with increasingly fine $p$ -adic level structure to acquire a geometric limit.

References

Peter Scholze, “Perfectoid spaces,” Publications Mathématiques de l'IHÉS 116 (2012), 245–313. arXiv.
Peter Scholze and Jared Weinstein, Berkeley Lectures on $p$ -adic Geometry, Annals of Mathematics Studies 207, Princeton University Press, 2020, Chapters 6–7.