Reductions towards a characteristic free proof of the Canonical Element Theorem

TL;DR

This paper reduces Hochster's Canonical Element Conjecture to a localization problem in a characteristic-free manner, proposing new variants and linking them to big Cohen-Macaulay modules to seek a characteristic-free proof.

Contribution

It introduces a new variant of the Canonical Element Theorem and connects its characteristic-free proof to the existence of Cohen-Macaulay complexes and modules.

Findings

Reduction of CET to a localization problem

Validation of a new variant of CET

Linking big Cohen-Macaulay modules to characteristic-free proofs

Abstract

We reduce Hochster's Canonical Element Conjecture (theorem since 2016) to a localization problem in a characteristic free way. We prove the validity of a new variant of the Canonical Element Theorem (CET) and explain how a characteristic free deduction of the new variant from the original CET would provide us with a characteristic free proof of the CET. We also show that the Balanced Big Cohen-Macaulay Module Theorem can be settled by a characteristic free proof if the big Cohen-Macaulayness of Hochster's modification module can be deduced from the existence of a maximal Cohen-Macaulay complex in a characteristic-free way.