A length formula for the multiplicity of distinguished components of intersections

To an arbitrary ideal I in a local ring image one can associate a multiplicity j(I,A) that generalizes the classical Hilbert–Samuel multiplicity of an image-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/(x1,…,xd−1)+xdI where dcolon, equalsdim A and x1,…,xd are sufficiently generic elements of I. This generalizes the classical length formula for image-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 image, where jacH is the Jacobian ideal generated by the partial derivatives of a defining equation of H.