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Perfectoid spaces

Peter Scholze

Abstract

We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings' almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. We deduce the weight-monodromy conjecture in certain cases by reduction to equal characteristic.

Perfectoid spaces

Abstract

We introduce a certain class of so-called perfectoid rings and spaces, which give a natural framework for Faltings' almost purity theorem, and for which there is a natural tilting operation which exchanges characteristic 0 and characteristic p. We deduce the weight-monodromy conjecture in certain cases by reduction to equal characteristic.

Paper Structure

This paper contains 9 sections, 92 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Adic spaces
  3. Perfectoid fields
  4. Almost mathematics
  5. Perfectoid algebras
  6. Perfectoid spaces: Analytic topology
  7. Perfectoid spaces: Etale topology
  8. An example: Toric varieties
  9. The weight-monodromy conjecture

Key Result

Theorem 1.1

The absolute Galois groups of $\mathbb{Q}_p(p^{1/p^\infty})$ and $\mathbb{F}_p((t))$ are canonically isomorphic.

Theorems & Definitions (143)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Claim 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 133 more