  For an associative ring $R$, the projective level of a complex $F$ is the
smallest number of mapping cones needed to build $F$ from projective
$R$-modules. We establish lower bounds for the projective level of $F$ in terms
of the vanishing of homology of $F$. We then use these bounds to derive a new
version of The New Intersection Theorem for level when $R$ is a commutative
Noetherian local ring.
