On the direct summand conjecture and its derived variant

TL;DR

This paper offers a simplified proof of Hochster's direct summand conjecture and its derived variant by utilizing a quantitative form of Scholze's Hebbarkeitssatz, avoiding the more complex perfectoid Abhyankar lemma.

Contribution

It introduces a quicker proof method for Hochster's conjecture using a new quantitative approach to the Riemann extension theorem for perfectoid spaces.

Findings

Simplified proof of Hochster's conjecture

Extension of the method to derived variants

Avoidance of the perfectoid Abhyankar lemma

Abstract

Andr\'e recently gave a beautiful proof of Hochster's direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where "discriminants" acquire all $p$-power roots. In this paper, we explain a quicker proof of Hochster's conjecture that circumvents the perfectoid Abhyankar lemma; instead, we prove and use a quantitative form of Scholze's Hebbarkeitssatz (the Riemann extension theorem) for perfectoid spaces. The same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de Jong.