La conjecture du facteur direct (d'apr\`es Y.Andr\'e et B.Bhatt)

TL;DR

This paper reviews Y. André's work on the direct summand conjecture and subsequent advances by André and Bhatt on big Cohen-Macaulay algebras, highlighting key theoretical developments in commutative algebra.

Contribution

It summarizes significant progress in the direct summand conjecture and big Cohen-Macaulay algebras, integrating recent breakthroughs and their implications.

Findings

Proof of the direct summand conjecture in mixed characteristic

Construction of big Cohen-Macaulay algebras in new settings

Resolution of longstanding open problems in commutative algebra

Abstract

This is a Bourbaki report on the work of Y. Andr\'e on the direct summand conjecture, and subsequent developments by Andr\'e and Bhatt on big Cohen-Macaulay algebras.