Perfectoid spaces and the homological conjectures

TL;DR

This survey discusses recent progress in commutative algebra in mixed characteristic using perfectoid spaces, including the author's proof of the direct summand conjecture and advances on homological conjectures.

Contribution

It explains the application of perfectoid space techniques to solve longstanding homological conjectures in mixed characteristic.

Findings

Proof of the direct summand conjecture using perfectoid spaces

Significant progress on homological conjectures in mixed characteristic

Clarification of techniques involving perfectoid spaces in algebra

Abstract

This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short account of the author's proof of the direct summand conjecture are included. One then portrays the progresses made with these (and related) techniques on the so-called homological conjectures.