Lower bounds on projective levels of complexes

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.