A study of perfectoid rings via Galois cohomology

TL;DR

This paper explores the properties of perfectoid rings and their tilts, clarifying their algebraic and homological aspects within $p$-adic Hodge theory, building on foundational work by Faltings and Scholze.

Contribution

It provides new results on the ring-theoretic and homological properties of tilts of perfectoid ring extensions, advancing understanding in $p$-adic Hodge theory.

Findings

Clarified homological properties of tilts of perfectoid rings

Established new ring-theoretic results related to perfectoid extensions

Enhanced understanding of perfectoid rings in the context of $p$-adic Hodge theory

Abstract

In his foundational study of $p$-adic Hodge theory, Faltings introduced the method of almost \'etale extensions to establish fundamental 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.