Descent of Flatness and the Direct Summand Conjecture

TL;DR

This paper proves a key algebraic theorem about complete regular local rings and their integral closures, establishing descent of flatness and the direct summand conjecture without using almost mathematics, applicable in all characteristics.

Contribution

It provides a new proof of the direct summand conjecture and descent of flatness for integral extensions, avoiding almost mathematics and working universally across characteristics.

Findings

Proves non-vanishing of Hom for integral closures in complete regular local rings.

Establishes descent of flatness for integral extensions of Noetherian rings.

Validates the direct summand conjecture in all characteristics.

Abstract

In the central theorem of this article we prove the following: if $R$ is a complete regular local ring and $B$ is the integral closure of $R$ in the algebraic closure of the fraction field of $R$, then $\Hom_R(B, R) \neq 0$. Our proof of this theorem does not involve almost mathematics and it works for all characteristics. As consequences we derive the validity of a) descent of flatness for integral extensions of Noetherian rings and b) direct summand conjecture due to M. Hochster.