  It is proved, as was conjectured by Eisenbud-Koh-Stillman, that for a
finitely generated graded module $M$ over the symmetric algebra $S(V)$, if the
Koszul group ${\cal K}_{p,0}(M,V)\ne 0$, then the set of rank 1 relations in
$M_0\otimes V$ has dimension $\ge p$. The method is by using ``exterior minors"
to study syzygies of an ideal derived from the Koszul class in the exterior
algebra. As a consequence, a conjecture of Lazarsfeld and myself that a set $Z$
of $2r+1-p$ points in ${\bf P}^r$ for which property $N_p$ fails has a subset
$Z'$ of at least $2{\rm dim}(L) +2-p$ points lying on a linear space $L$ such
that property $N_p$ fails for $Z'$.
