---
tag: Q9Y4CJE1X5
kind: concept
name: Perfectoid space
msc: [14G45, 14G22, 12J25]
sources:
  - url: https://en.wikipedia.org/wiki/Perfectoid_space
same_as:
  wikidata: Q17075276
  wikipedia: Perfectoid space
---

## Summary

A perfectoid space is an analytic space over a non-archimedean field of residue characteristic $p$.
It is so ramified that the Frobenius map is surjective modulo $p$.
Peter Scholze introduced perfectoid spaces in 2012.
Their key feature is tilting: perfectoid spaces in mixed characteristic correspond to perfectoid spaces in characteristic $p$.
This lets one transfer problems between the two worlds.
They have become a basic tool of $p$-adic geometry and the arithmetic of [[Galois representations|JQEYEQJY2G]].

## Explanations

### L1 human

The $p$-adic numbers and the field of Laurent series over $\F_p$ look very different, since one has characteristic $0$ and the other characteristic $p$.
Adjoin all $p$-power roots of $p$ to the first, all $p$-power roots of the variable to the second, and complete both.
The two resulting fields have equivalent Galois theories.
Perfectoid spaces extend this surprising match from fields to geometry.
A hard question in characteristic $0$ can then be moved to characteristic $p$, where the Frobenius map gives extra tools.