La conjecture du facteur direct

TL;DR

This paper proposes a proof of the direct factor conjecture using perfectoid theory, focusing on unramified regular local rings of mixed characteristic and employing Kummer extensions and the Abhyankar lemma.

Contribution

It introduces a novel proof of the direct factor conjecture leveraging perfectoid techniques and Kummer extensions in mixed characteristic settings.

Findings

Proof of the direct factor conjecture in mixed characteristic

Application of perfectoid Abhyankar lemma to local rings

Analysis of Kummer extensions via thickening techniques

Abstract

Building on his reduction to the case of an unramified complete regular local ring R of mixed characteristic, we propose a proof in the framework of P. Scholze's perfectoid theory. The main ingredients are the perfectoid "Abhyankar lemma" and an analysis of Kummer extensions of R by a thickening technique.