Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A study of perfectoid rings via Galois cohomology

Ryo Kinouchi, Kazuma Shimomoto

Abstract

In his foundational study of $p$-adic Hodge theory, Faltings introduced the method of almost étale extensions to establish fundmental comparison results of various $p$-adic cohomology theories. Scholze introduced the tilting operations to study algebraic objects arising from $p$-adic Hodge theory in mixed characteristic via the Frobenius map. In this article, we prove a few results which clarify certain ring-theoretic or homological properties of the tilt of an extension between perfectoid rings treated in the construction of big Cohen-Macaulay algebras.

A study of perfectoid rings via Galois cohomology

Abstract

In his foundational study of -adic Hodge theory, Faltings introduced the method of almost étale extensions to establish fundmental comparison results of various -adic cohomology theories. Scholze introduced the tilting operations to study algebraic objects arising from -adic Hodge theory in mixed characteristic via the Frobenius map. In this article, we prove a few results which clarify certain ring-theoretic or homological properties of the tilt of an extension between perfectoid rings treated in the construction of big Cohen-Macaulay algebras.
Paper Structure (3 sections, 12 theorems, 32 equations)

This paper contains 3 sections, 12 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Main theorems

Key Result

Theorem 1.1

Let the notation be as above. Then $R_{\infty} \to R_{\infty,p}$ is an inductive colimit of $(p)^{1/p^\infty}$-almost finite étale extensions and the $p$-adic completion of $R_{\infty,p}$ is perfectoid. Moreover, the natural map of Galois cohomology groups is a $(p)^{1/p^\infty}$-almost isomorphism for every $i \ge 0$.

Theorems & Definitions (28)

  • Theorem 1.1: Faltings
  • Theorem 1.2: Bhatt, Hochster-Huneke
  • Corollary 1.3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Almost $G$-Galois cover
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 18 more