Almost direct summands

TL;DR

This paper demonstrates how recent advances in p-adic Hodge theory can be used to prove new cases of Hochster's direct summand conjecture, especially in characteristic p ramification scenarios.

Contribution

It introduces a novel approach linking p-adic Hodge theory to the direct summand conjecture, expanding the known cases where the conjecture holds.

Findings

New cases of Hochster's conjecture are established in characteristic p.

Fundamental p-adic Hodge theorems are applied to commutative algebra problems.

The approach simplifies proofs of certain direct summand cases.

Abstract

In this short note, we point out how some new cases of Hochster's direct summand conjecture can be deduced from fundamental theorems in p-adic Hodge theory due to Faltings. The cases tackled include the ones when the ramification of the map being considered lies entirely in characteristic p.