  Let $I$ be an ideal of height $d$ in a regular local ring $(R,m,k=R/m)$ of
dimension $n$ and let $\Omega$ denote the canonical module of $R/I$. In this
paper we first prove the equivalence of the following: the non-vanishing of the
edge homomorphism $\eta_d: \ext{R}{n-d}{k,\Omega} \rightarrow \ext{R}{n}{k,R}$,
the validity of the order ideal conjecture for regular local rings, and the
validity of the monomial conjecture for all local rings. Next we prove several
special cases of the order ideal conjecture/monomial conjecture.
