The monomial conjecture and order ideals

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 of certain syzygies of canonical modules of normal domains possessing free summands. We also prove some special cases of this observation.