Perfectoid spaces
Overview
A class of Banach algebras and their spaces, introduced by Scholze (2012). Tilting relates mixed-characteristic and equal-characteristic objects; key to the proof of the weight-monodromy conjecture.
Era: 2011–2012
Proposed at: Bonn, Germany
Formal structure
Domain: Perfectoid ring R; complete, Tate; Frobenius surjective mod p
R^♭ = lim_{←φ} R/p; tilting: R ↦ R^♭ (char p ↦ char 0)
Constraints: Frobenius: R/p → R/p surjective; ϖ^p | p for pseudo-uniformizer ϖ
Objective: Weight-monodromy; p-adic Hodge theory; Fargues–Fontaine