A length formula for the multiplicity of distinguished components of intersections

To an arbitrary ideal I in a local ring (A,m) one can associate a multiplicity j(I,A) that generalizes the classical Hilbert-Samuel multiplicity of an m-primary ideal and which plays an important role in intersection theory. If the ideal is strongly Cohen-Macaulay in A and satisfies a suitable Artin-Nagata condition then our main result states that j(I, M) is given by the length of. I/(x(1),...,x(d-1)) + x(d)I where d:=dimA and x(1),...,x(d) are sufficiently generic elements of I. This generalizes the classical length formula for m-primary ideals in Cohen-Macaulay rings. Applying this to an hypersurface H in the affine space we show for instance that an irreducible component C of codimension c of the singular set of H appears in the self-intersection cycle Hc+1 with multiplicity e(jac(H,C), O-H,O-C), where jac(H) is the Jacobian ideal generated by the partial derivatives of a defining equation of H. (C) 2001 Elsevier Science B.V. All rights reserved.