  We prove a projection formula, expressing a relative Buchsbaum--Rim
multiplicity in terms of corresponding ones over a module-finite algebra of
pure degree, generalizing an old formula for the ordinary (Samuel)
multiplicity. Our proof is simple in spirit: after the multiplicities are
expressed as sums of intersection numbers, the desired formula results from two
projection formulas, one for cycles and another for Chern classes. Similarly,
but without using any projection formula, we prove an expansion formula,
generalizing the additivity formula for the ordinary multiplicity, a case of
the associativity formula.
