The Monomial Conjecture and Order Ideals

TL;DR

This paper links a specific case of the order ideal conjecture to Hochster's monomial conjecture, providing conditions involving syzygies of canonical modules and proving some special cases.

Contribution

It establishes a new connection between the order ideal conjecture and the monomial conjecture, with criteria involving syzygies of canonical modules in normal domains.

Findings

A special case of the order ideal conjecture implies the monomial conjecture.

Necessary and sufficient conditions are derived involving syzygies of canonical modules.

Some special cases of these conditions are proven.

Abstract

In this article first we prove that a special case of the order ideal conjecture, originating from the work of Evans and Griffith in equicharacteristic, implies the monomial conjecture due to M. Hochster. We derive a necessary and sufficient condition for the validity of this special case in terms certain syzygis of canonical modules of normal domains possessing free summands. We also prove some special cases of this observation.