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.